continued fractions

Continued Fractions: From Analytic Number Theory to Constructive Approximation

A Volume in Honor of L. J. Lange

Continued Fractions: From Analytic Number Theory to Constructive Approximation

May 20-23, 1998 University of Missouri-Columbia

Bruce C. Berndt Fritz Gesztesy

Editors

American Mathematical Society Providence, Rhode Island

Editorial Board

Dennis DeTurck, managing editor

Andreas Blass Andy R. Magid Michael Vogelius

This volume contains t h e contributions t o t h e international conference "Continued Fractions: From Analytic Number Theory t o Constructive Approximation", held at t h e University of Missouri-Columbia on May 20-23, 1998.

1991 Mathematics Subject Classzjication. Primary 42C05, 30B70, 30E05, 40A15; Secondary 11-XX, 33-XX, 41-XX.

L i b r a r y of Congress Cataloging-in-Publication D a t a

Continued fractions : from analytic number theory to constructive approximation : a volume in honor of L. J. Lange : continued fractions, from analytic number theory t o constructive approximation, May 20-23, 1998, University of Missouri / Bruce C. Berndt, Fritz Gesztesy, editors.

p. cm. - (Contemporary mathematics, ISSN 0271-4132 ; v. 236) Includes bibliographical references. ISBN 0-8218-1200-9 (alk. paper) 1. Continued fractions-Congresses. I. Lange, L. J. (Leo Jerome), 1928- . 11. Berndt,

Bruce C., 1939- . 111. Gesztesy, Fritz, 1953- . IV. Series: Contemporary mathematics (Ameri- can Mathematical Society) ; v. 236. QA295.C663 1999 515l.243-dc21 99-30750

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Contents

Preface and Dedication

List of Participants

ix

xvi

Continued fractions and orthogonal polynomials R. ASKEY

The problems submitted by Ramanujan to the Journal of the Indian Mathematical Society B. C. BERNDT, Y.-S. CHOI, AND S.-Y. KANG

Some examples of moment preserving approximation B. BOJANOV AND A. SRI RANGA

Relations between certain symmetric strong Stieltjes distributions C. F. BRACCIALI

Estimates of the rate of convergence for certain quadrature formulas on the half-line A. BULTHEEL, C. D~Az-MENDOZA, P. GONZALEZ-VERA, AND

R. ORIVE

Wavelets by orthogonal rational kernels A. BULTHEEL AND P. GONZALEZ-VERA

On the explicit evaluations of the Rogers-Ramanujan continued fraction H. H. CHAN AND V. TAN

Absence of phase transitions in modified two-component plasmas: The analytic theory of continued fractions in statistical mechanics D. CHELST

Some continued fractions related to elliptic functions M. E. H. ISMAIL AND D. R. MASSON

Asymptotics of Stieltjes continued fraction coefficients and applications to Whittaker functions W. B. JONES AND G. SHEN

A generalization of Van Vleck's theorem and more on complex continued fractions L. J. LANGE

vii

. . . vl11 CONTENTS

Convergence of interpolating Laurent polynomials on an annulus X. LI

Convergence criteria for continued fractions K(a,/l) based on value sets L. LORENTZEN

Strong Stieltjes moment problems 0. N J ~ S T A D

Weak asymptotics of orthogonal polynomials on the support of the measure of orthogonality and considerations on functions of the second kind F. PEHERSTORFER AND R. STEINBAUER

Trees of approximation constants S. PERRINE

Continued fractions and Schrodinger evolution I. RODNIANSKI

Multiple orthogonal polynomials, irrationality and transcendence W. VAN ASSCHE

Reduction of continued fractions of formal power series A. J. VAN DER POORTEN

Some observations in frequency analysis H. WAADELAND

Some properties of Hermite-Padk approximants to eZ F. WIELONSKY

Preface and Dedication

This volume contains the refereed contributions of the international conference "Continued Fractions: From Analytic Number Theory to Constructive Approxi- mation", held at the University of Missouri-Columbia on May 20-23, 1998. The meeting also celebrated Jerry Lange's seventieth birthday and marked his retire- ment from the University of Missouri. It is a great pleasure to dedicate this volume to Jerry in recognition of his distinguished service and long-lasting impact on the Mathematics Department at MU.

In spite of their long tradition, continued fractions (whose general definition appears to go back to the book "Liber Abaci" of Leonardo of Pisa, also called Fi- bonacci, written around 1202) remain an active area of research in a large number of fields ranging from pure mathematics to mathematical physics and approximation theory. The principal purpose of this conference was to focus on continued fractions as a common interdisciplinary theme bridging gaps between these fields. As a consequence, the lectures at this conference and the corresponding contributions to this volume reflect the wide range of applicability of continued fractions in mathematics and the applied sciences.

More specifically, recurrence relations for orthogonal polynomials and their relations to continued fractions, appealing to a well-known theorem of Markoff, are studied in Askey's contribution. Orthogonal Laurent polynomials and questions of (in)determinacy of the strong Stieltjes moment problem are considered by Njktad. Strong Stieltjes distributions, orthogonal Laurent polynomials and related continued fractions appear in Bracciali's article. Generating functions of orthogonal Laurent polynomials, associated kernel polynomials, and moment-preserving approximations are treated by Bojanov and Sri Ranga. Convergence properties of Laurent polynomials that interpolate functions on the boundary of a circular annulus are investigated by Li. Compact perturbations of reflection coefficients and the resulting weak asymptotics of orthogonal polynomials on the support of the measure of orthogonality and asymptotics of the corresponding functions of the second kind are studied by Peherstorfer and Steinbauer.

Convergence of Stieltjes continued fractions and their asymptotic speed of convergence as a result of analyzing the asymptotic behavior of the corresponding expansion coefficients, applicable to Whittaker functions, are considered by Jones and Shen. Various convergence theorems including a new and constructive proof of an extension of Van Vleck's convergence theorem for continued fractions, including a sharp truncation error formula, are derived in Lange's contribution. A very extensive treatment of convergence criteria, including substantial progress in proving a conjecture by Lorentzen and Ruscheweyh, as well as a simplified approach to Cordova's extension of the classical parabola theorem, are provided by Lorentzen.

ix

x PREFACE AND DEDICATION

A thorough survey of all 58 problems submitted by Ramanujan to the Journal of the Indian Mathematical Society between 1911 and 1919, detailing their subsequent influence on various directions of research since 1927, is provided by Berndt, Choi, and Kang. Recent progress made in explicit evaluations of the Rogers-Ramanujan continued fractions using modular equations and class invariants, are discussed by Chan and Tan. Continued fractions of Stieltjes and Rogers associated with elliptic functions and solutions of the corresponding difference equations are studied by Ismail and Masson.

A sequence of approximation constants converging towards 113 by lower values associated with the Diophantine equation x 2 + y2 + z2 = 3 x y z + 22 is developed by Perrine. A principle relating the reduction mod p of a continued fraction expansion to the original expansion in characteristic zero and illustrations involving classical Abelian integrals and hyperelliptic curves are discussed by van der Poorten. Multiple orthogonal polynomials, a notion closely related to Hermite-Pad6 rational approximation of a system of Markoff functions, and constructive proofs of the irrationality of C(3) and transcendence of certain real numbers (applicable to e ) are presented by Van Assche. Convergence questions and the location of zeros of Hermite-Pad6 approximants to eZ are considered by Wielonsky.

Orthogonal reproducing rational kernels are used to construct rational wavelets on the unit circle with respect to arbitrary positive measures by Bultheel and GonzAlez-Vera. The rate of convergence for n-point Gauss-type quadratic integrals on the half-line is estimated by Bultheel, Diaz-Mendoza, GonzAlez-Vera, and Orive. A new approach to frequency analysis for trigonometric signals, circumventing the associated problem of vanishing Toplitz determinants, is presented by Waadeland.

The absence of pressure-dependent phase transitions in modified two-component plasmas, extending Lenard's original approach based on continued fractions, is shown by Chelst,. Spatial regularity properties of the fundamental solution (propagator) of the time-dependent Schrodinger equation on the circle in the presence of a complex-valued potential in terms of Besov spaces, depending on arithmetic properties of the time parameter, are derived by Rodnianski.

Naturally, this event required support by a number of individuals. We thank the remaining organizing committee members, Nigel Kalton, Igor Verbitsky, and especially Mark Ashbaugh for their assistance in preparing this conference. Special thanks are due to all staff members at the Mathematics Department, especially to Bridget Kelley and some of our graduate students for their great efforts on behalf of this conference. Moreover, we also thank Helge Holden and Karl Unterkofler for their help over the past two years. Holden generously shared his considerable ex- perience in organizing events of this type; Unterkofler handled all technical aspects in producing this proceedings volume and provided invaluable assistance to several contributors and the editors.

Finally, we gratefully acknowledge financial support for this conference from a variety of sources, including the US National Science Foundation (NSF DMS 9729700), the University of Missouri Research Board (RB98-012), The College of Arts & Science, The Office of Research, and the Department of Mathematics at MU. Special thanks are due to Elias Saab, who early on enthusiastically supported the idea of such an event and, in his position as Chair of the Mathematics Department, initially guaranteed a major portion of funding for this project.

L. J. Lange

L. J . Lange on His 70th Birthday

Leo Jerome Lange (Jerry) was the first of five children born to Leo and Clara Lange on August 29, 1928, in New Rockford, North Dakota. For most of his boyhood years, Jerry was raised on a farm about eleven miles from New Rockford. This farm was homesteaded by his grandparents, and it was the birthplace of his father. There he learned the value of hard work and experienced some of the adversity brought on by the aftermath of the Great Depression and the dust bowl era of the thirties.

Jerry's first eight years in school were spent in a one-room country school house. He was fortunate to have several inspiring teachers in this school and recalls reading most of the books (including the encyclopedias) in its limited library.

Jerry spent his four high school years, 1942-1946, at Sacred Heart Academy (SHA) (now Schanley High School) in Fargo, North Dakota. To continue to attend this premier high school (about 175 miles from the family home) he lived and worked in a hospital in Fargo during his last three years at the school. To this day he much appreciates the education he received from excellent dedicated teachers at SHA. He has an especially fond memory of a nun who had the uncanny ability of getting teen-age boys enthused about presenting proofs of propositions on the blackboard.

Upon graduation from high school at the rank of Salutatorian of his class, Jerry now faced the problems of how to finance his desired college education and what to do about the fact that he was about to be caught up in the post World War I1 military draft. The possible educational benefits of the WW I1 GI Bill influenced him to enlist in the US Army on September 30, 1946 for an eighteen month term. He served as a medic for more than a year in Berlin, Germany in the 3rd. Battalion, 16th Infantry Regiment, of the 1st. Division, and obtained the final rank of Corporal.

Through the benefits of the GI Bill, by working in the summer, and with some extra "travel money" from his parents, Jerry was able to attend Regis College (now Regis University) in Denver, Colorado. He started his training there in the fall of 1948 and eventually decided to major in Mathematics and minor in Physics and Philosophy. He is very appreciative of the challenges that were presented to him and the attitudes that were fostered in him by his professors at Regis. Jerry received the Most Outstanding Senior Award from Regis, and graduated with a BS degree in the spring of 1952.

Jerry had been encouraged by his mathematics teachers at Regis to pursue an advanced degree in mathematics, and in August of 1952 he went to the University of Colorado-Boulder (CU) to seek admission as a graduate student in mathematics. Burton W. Jones, who was then Chair of the Mathematics Department at CU,

xii

L. .I. LANGE O N HIS 70TH BIRTHDAY . . . X l l l

accepted him immediately. Jerry began his teaching career in the winter semester of 1953 when he received a Teaching Assistantship in Mathematics.

In June of 1955, he married Geraldine Ryan who was teaching music in the Denver public school system. They lived in Denver and Jerry commuted to Boulder daily to continue his study and teaching at CU. By the end of the summer of 1955 he had completed his Masters Thesis on "Some Problems in Interpolation" and all of the requirements for the MA degree. His Masters Thesis advisor was Kurt A. Hirsch of the University of London, and the second reader for the work was S. Chowla.

In 1956 Geraldine and Jerry had started a family, so she chose to relinquish her teaching position in Denver. Jerry accepted a classified position as a mathematician at the National Bureau of Standards, Boulder Laboratories (NBS) in July of 1956. His position at NBS was located in the Low Frequency and Very Low Frequency Section of the Radio Communication and Systems Division. For the next two years he continued to take graduate courses at CU while working full-time at NBS, making up the time he spent in class by working on weekends. Jerry did most of the analyses and prepared most of the technical materials that were used by the US delegation to the International Technical Discussions on Detection of Nuclear Tests held in Geneva, Switzerland in the summer of 1958. For his work at NBS, Jerry received two merit awards from the United States Department of Commerce, one in 1959 and one in 1960.

In the fall of 1958, through his Ph.D. advisor Wolfgang J . Thron, Jerry was awarded a two-year AFOSR grant for graduate research. This allowed him to convert his position at NBS to half-time, so he could spend his mornings at CU attending classes and doing research for his doctoral thesis. Jerry was the first of Thron's many Ph.D. students, and he was awarded the Ph.D. degree from CU in the summer of 1960. His doctoral thesis consists of three parts and is entitled "Divergence, Convergence, and Speed of Convergence of Continued Fractions 1 + K(a,/l)". The Convergence part appears in his joint 1960 publication with Thron in Mathematische Zeitschrifl, and t,he other two parts have played significant roles in some of Jerry's later publications. Upon receiving t'he Ph.D., he was offered a large increase in civil service rank if he would continue his work at NBS. However, he chose to resign at NBS in August, 1960 so that he could accept an academic position at the University of Missouri-Columbia (MU).

Jerry joined the faculty of the Department of Mathematics at MU on September 1, 1960. He has spent his entire academic career at this institution. He served as Chair of the Department during the years 1988-91 and during the summers of 1966, 1968, and 1969. He also served in the position of Associate Chair during the years 1968-69, 1977-78, and 1978-79. His teaching record shows that he taught in the neighborhood of one hundred and ninety classes at MU, most of which were at the calculus level and beyond. He was a pioneer in the use of the computer as a teaching aid in calculus, and for three decades he was the principal creator, teacher, and promoter of complex analysis courses in the Department. Jerry gave unselfishly of his time to numerous important Departmental and Campus personnel, academic, and policy committees. He is especially proud of his role in the Departmental Planning Committee that had much to do with charting the path of the Department in the last decade.

xiv I. .J L A N G E O N HIS 70TH BIRTHDAY

Jerry's principal field of research has been the analytic theory of continued fractions. He is an internationally recognized expert on the convergence of continued fractions and their applications to function representation in the complex domain. In each of his papers on continued fractions he has solved one or more, usually long-standing, problems in the field. Space limitations prohibit a discussion of Jerry's research in detail here. but the majority of his results are fundamental, have wide-reaching significance, and are cited often by other researchers. His Uni- form Twin Liina~on Theorem for continued fractions K ( a , / l ) . which he proved in his 1966 paper in the Illznozs Journal of Mathematzcs, has turned out to be one of the most significant results in convergence region theory. Recently. in the Campznas Proceedzngs. he settled a twenty-eight-year-old conjecture of Jones and Thron by proving that almost all twin convergence regions for K ( a , / l ) generated by either disk-disk, disk-halfplane, or disk-complement of disk mappings are embeddable in those of his Lima~on Theorem. In two papers in the early 1980's, Jerry developed the theory of 6-fractions. His &fraction work solved a number of problems dealing with the association of continued fractions and power series for analytic functions. Others have used his 6-fractions to approximate solutions of Riccati differential equations, and Jerry applied them to zero location and stability problems in one of his 1986 papers. I11 his 1986 work with Kalton, many outstanding problems in the area of equimodular limit-periodic continued fractions were solved. More than ten years later. some of this work was used to disprove an assertion of Ramanujan and now appears in Berndt's Ramanujan's Notebooks, Part V. The Worpitzky and Transcendental strips in his 1994 paper on strip convergence regions for continued fractions amount to the first known best convergence regions that are distortions of the famous parabolic regions and have inspired others to seek generalizations of them. In his 1995 paper in Constructzve Approxzmatzon, Jerry completely settles the Oval Theorem for continued fractions and proves an earlier result of others false by actually showing that these ovals are embeddable in the parabolas of the Uniform Parabola Theorem. His Uniform Twin Parabola Theorem of 1994 in the Journal of Mathematzcal Analyszs and Applzcatzons generalizes work of Thron fifty years earlier in several important ways. Uniformity and speed of convergence are obtained, and only the even elements need to be bounded. His work on Van Vleck fractions K ( l / b , , ) in this publication should prove to be quite valuable in future research on complex continued fractions of this type. Jerry has participated by in- vitation in many research workshops and conferences in his field, including Boulder and Redstone. Colorado. Pitlochry and Aviemore, Scotland, Loen and Trondheim, Norway. and Campinas. Brazil.

Geraldine and Jerry have four children (two sons and two daughters) and nine grandchildren. Jerry retired on September 1, 1998, with the status of Professor Emeritus after thirty-eight years of service at MU. He is looking forward to enjoying mathematics, his family, and his other interests for a long time to come.

We wish the very best to Jerry in his future personal and professional endeavors.

Bruce C. Berndt Fritz Gesztesy

Publications of L. J. Lange

[I] Ionospheric reflection coefficients at VLF from sferics measurements, Geofisica Pura Applicata 38 (1957) 147-153 (with A. G. Jean and J. R. Wait).

[2] Some characteristics of VLF propagation using atmospheric waveforms, Recent Advances in Atmospheric Electrzcity, pp. 609-617, Permagon Press, London-New York-Paris-Los Angeles, 1959 (with W. L. Taylor).

(31 A two-parameter family of best twin convergence regions for continued fractions, Math. Z. 73 (1960) 295-311 (with W. J. Thron).

[4] On a family of twin convergence regions for continued fractions, Illinois J. Math. 10 (1966), 97-108.

[5] A simple irrationality proof for nth roots of positive integers, Math. Mag. 42 (1969) 242-244. [6] &fraction expansions of analytic functions, Analytic Theory of Continued Fractions (Loen,

1981), pp. 152-175, Lecture Notes in Math. 932, Springer, Berlin-New York, 1982. [7] &fraction expansions of analytic functions, SIAM J. Math. Anal. 14 (1983) 323-368. [8] Equimodular limit periodic continued fractions, Analytic Theory of Contznued Fractions, I I

(Pitlochry/Aviemore, 1985), pp. 159-219, Lecture Notes In Math. 1199, Springer, Berlin-New York, 1986 (with N. J. Kalton).

[9] Continued fraction applications to zero location, Analytic Theory of Continued Fractions, I1 (Pitlochry/Aviemore, 1985), pp. 220-262, Lecture Notes In Math. 1199, Springer, Berlin-New York, 1986.

[lo] Continued fraction representations for functions related to the gamma function, Continued Fractions and Orthogonal Functions (Loen, 1992), pp. 233-279, Lecture Notes in Pure and Appl. Math. 154, Dekker, New York, 1994.

[ll] Strip convergence regions for continued fractions, Contznued Fractions and Orthogonal Func- tions (Loen, 1992), pp. 211-231, Lecture Notes in Pure and Appl. Math. 154, Dekker, New York, 1994.

[12] A uniform twin parabola convergence theorem for continued fractions, J. Math. Anal. Appl. 188 (1994) 985-998.

[13] Convergence region inclusion theorems for continued fractions K(a,/l), Constr. Approx. 11 (1995) 321-329.

1141 Uniformity and speed of convergence of complex continued fractions K(a,,/l) , Orthogonal Functions, Moment Theory, and Continued Fractions (Campinas, 1996), pp. 275-326, Lec- ture Notes in Pure and Appl. Math. 199, Dekker, New York, 1998.

[15] Convergence regions with bounded convex complements for continued fractions K(l/bn), J. Comp. Appl. Math. 105 (1999), pp. 355-366.

[16] A generalization of Van Vleck's theorem and more on complex continued fractions, Continued Fractions: From Analytic Number Theory to Constructive Approximation, Contemporary Mathematics 236 (1999), pp. 179-192.

[17] An elegant continued fraction for T , Amer. Math. Monthly 106 (1999), pp. 456-458.

A List of Participants

M. Ashbaugh, University of Missouri, Columbia E-Mail: [email protected]

R. Askey, University of Wisconsin, Madison E-Mail: [email protected]

J. Beem, University of Missouri, Columbia E-Mail: [email protected]

R. Benguria, P. Universidad Cat6lica de Chile, Santiago, Chile E-Mail: [email protected]

B.C. Berndt, University of Illinois, Urbana-Champaign E-Mail: [email protected]

D. Bowman, University of Illinois, Urbana-Champaign E-Mail: [email protected]

K. N. Boyadzhiev, Ohio Northern University, Ada E-Mail: [email protected]

C. F. Bracciali, State University of Siio Paulo, Brazil, Current address: Mathematical Institute, University of St Andrews, Scotland E-Mail: [email protected]

A. Bultheel, Catholic University of Leuven, Belgium E-Mail: [email protected]

H. H. Chan, National University of Singapore, Singapore E-Mail: [email protected]

D. Chelst, Rutgers, New Jersey E-Mail: [email protected]

C. Chicone, University of Missouri, Columbia E-Mail: [email protected]

G. Choi, University of Illinois, Urbana-Champaign E-Mail: [email protected]

Y .-S. Choi, University of Illinois, Urbana-Champaign E-Mail: [email protected]

S. Clark, University of Missouri, Rolla E-Mail: [email protected]

J. Conlon, University of Michigan, Ann Arbor E-Mail: [email protected]

R. Crownover, University of Missouri, Columbia E-Mail: [email protected]

P. E. Ehrlich, University of Florida, Gainesville E-Mail: [email protected]

xvi

A LIST OF PARTICIPANI'S xvii

Z. Franco, Butler University, Indianapolis E-Mail: francoQthomas.butler.edu

F. Gesztesy, University of Missouri, Columbia E-Mail: [email protected]

P. GonzBlez-Vera, La Laguna University, Tenerife, Spain E-Mail: pglezQul1.e~

D. Hensley, Texas A&M University. College Station E-Mail: [email protected]

C. Hines. Institute for Defense Analyses, Fairview Hgts, IL E-Mail: cwhinesQapci.net

M. Ismail, University of South Florida, Tampa E-Mail: [email protected]

W. B. Jones, University of Colorado, Boulder E-Mail: [email protected]

I N. J. Kalton, IJniversity of Missouri, Columbia

E-Mail: nigelQmath.missouri.edu 1 L. J. Lange, University of Missouri, Columbia

E-Mail: jerryQmath.missouri.edu

1 Y. Latushkin, University of Missouri, Columbia E-Mail: [email protected]

X. Li, University of Central Florida, Orlando E-Mail: xliQpegasus.cc.ucf.edu

W.-C. Liaw, University of Illinois, Urbana-Champaign E-Mail: [email protected]

K. P. Litchfield, Farmington, Utah E-Mail: 1itchfieldkCQworldnet .att .net

L. Lorentzen, Norwegian University of Science and Technology, Trondheim, Norway E-Mail: [email protected]

P. Magnus, Ft. Collins, CO E-Mail: [email protected]

K. A. Makarov, University of Missouri, Columbia E-Mail: [email protected]~issouri.edu

B. Meller, P.U. Cat6lica de Chile, Santiago, Chile E-Mail: bmel1erQbohr.fis.puc.cl

M. Mitrea, University of Missouri, Columbia E-Mail: mariusQmath.missouri.edu

P. Nevai, Ohio State University, Columbus E-Mail: [email protected]

0 . Njbstad, Norwegian University of Science and Technology, Trondheim, Norway E-Mail: [email protected]

M. Pang, University of Missouri, Columbia E-Mail: [email protected]

S. Perrine, France Telecom, Metz, France E-Mail: [email protected]

T. Randolph, University of Missouri, Rolla E-Mail: [email protected]

A. S. Ranga, Universidade Estadual Paulista (UNESP), IBILCE, Brazil E-Mail: rangaQnimitz.dcce.ibilce.unesp.br

I. Rodnianski. Kansas State University, Manhattan E-Rlail: irod o3math.ksu.edu

F. Rsnning. S0r-Tr0ndelag College, Trondheim, Norway E-illail: [email protected]

C. Rousseau. The University of Memphis. Memphis, TN E-Rlail: rousseac0hermes.msci. memphis.edu

H.-J. Runckel, University of Ulm, Ulm, Germany E-Rlail: runckelQ!mathematik.uni-ulm.de

E. Saab. University of Rlissouri, Columbia E-illail: [email protected]

P. Saab, University of Missouri, Columbia E-Rlail: [email protected]

G. Shen, University of Colorado. Boulder E-illail: [email protected]

B. Simon, California 1nst;tute of Technology, Pasadena. CA E-Rlail: bsimonBcaltech.edu

V. Skarda. Brigham Young University, UT E-Mail: skardaCOmath.byu.edu

J. Sohn, University of Illinois, Urbana-Champaign E-Llail: [email protected]

K. Sollers, Cornell University. Ithaca. NY E-Rlail: [email protected]

S. H. Son. University of Illinois, Urbana-Champaign E-Mail: sonQmath.uiuc.edu

R. Steinbauer. Ohio State University. Columbus E-hlail: [email protected]

F. E. Su. Harvey Ivfudd College, Claremont. CA E-Llail: [email protected]

IV. Van Assche. Catholic University of Leuven, Belgium E-hlail: Walter.VanAsscheQwis.kuleuven.ac.be

I. Verbitsky. University of blissouri, Columbia E-Mail: [email protected]

H. Waadcland, Norwegian University of Science and Technology, Trondheirn. Norway E-Rlail: haakonu.aQmath.ntnu.no

B. Yeap, University of Illinois, Urbana-Champaign E-Mail: [email protected]

Z. Zhao, University of Rlissouri. Columbia E-Mail: zzhaoQ!math.missouri.edu

Contemporary blathrrr lat irs Vulnrnc 236. 1999

Continued fractions and orthogonal polynomials

Richard Askey

To Jerry Lange for carryzng o n the work wzth continued fractzons

A B S ~ I I A ~ T . Recurrence relations are very important when one studies orthogonal polynomials. Examples are given. including some where the recurrence relations relate to continued fractions.

1. Introduction

Gabor Szego wrote the following in the preface of his great book "Orthogonal Polynomials" . [34. p. v].

"The origins of the subject [orthogonal polynomials] are to be fourld in the investigation of a certain type of continued fractions. bearing the name of Stieltjes, . . . Despite the close relationship between continued fractions and the problem of moments. and notwithstanding recent important advances in this latter subject. continued fractions have been gradually abandoned as a starting point for the theory of orthogonal polynomials."

\Vhen Szego wrote his book sixty years ago. it was not clear what role was left for continued fractions. There are two dominant themes in Szego's book: the classical orthogonal polynomials of Jacobi. Laguerre and Hermite, and the general theory of polynomials orthogonal on a bounded interval, say [-I. 11. whose weight function is positive in the interior of this interval and does not vanish too rapidly as the variable approaches the endpoints of this interval. The classical orthogonal polynomials had been studied for about 150 years when Szego wrote [MI. with the work of Legendre and Laplace on spherical harmonics being one of the starting places. Gauss [14] developed Gaussian quadrature from continued fractions. How- ever. Jacobi's version [18] of this using orthogonality was much easier to motivate. and to extend to other measures.

Szego's theory of orthogonal polynomials on an interval. and the corresponding theory on the unit circle. were developed independently from continued fractions.

1991 Mathematzc Subject Classzficatzon. Primary 33C45. 33D45: Secondary 3OB7O. Key words and phrases. Orthogonal polynomials. corltiriued fractions, recurrcnce relations. Supported in part by a grant from the University of Wisconsin Graduate School.

2 RICHARD ASKEY

In the last sixty years there have been many new results obtained on orthogonal polynomials. In one of these areas, continued fractions play a major role. There are also very interesting combiriatorial connections with continued fractions and orthogonal polynornials. See Viennot [39].

2. The classical orthogonal polynomials

The traditional classical orthogonal polynomials. those of Jacobi. Laguerre and Hermite, have a number of properties which other orthogonal polynomials do not share. For example. if {p,, (r) ) is a set of orthogonal polynornials, and {p; +, ( T ) ) is also a set of orthogonal polynornials, then {p,,(x)} is orthogonal with respect to a beta, garnrna or normal distribution. Up to a linear transformation, {p,, (x)} is one of these polynornials. Jacobi, Laguerre or Hermite. They are defined by

which satisfies

which satisfies

(2.2a) LE ( x ) L ~ , ( x ) ~ ( ~ e - ~ d x = 0, m # n, a > -1,

or

Their derivative formulas are

d - H,, (J) = 2n H,, - 1 ( L ) . d r

The usual notation will be used for hypergeometric series:

with

There are extensions of these polynomials, first, as hypergeomeric series, and then as basic hypergeometric series. For hypergeometric series, the first extension

CONTINUED FRACTIONS AND ORTHOGONAL POLYNOMIALS 3

was of Legendre polynomials, the case with a = P = 0 of ( 2 . 1 ) . Thebychef [37] showed that

satisfies 2

(2.10) x Q,, ( x ; M)Q,, ( x : N ) = 0, rri # n 5 N. r=O

This is mentioned by Szego [34, Section 2.81. At the end of Chapter 2. Szego [34] mentions an extension of this result found by Markoff in his thesis [ 2 1 ] . The measure is qT, r = 0 , 1 , . . . , N.

In a later paper [38]. Tchebychef extended the polynomials defined in ( 2 . 9 ) to a more general set:

The orthogonality relatjion is

when rn, n = 0 , 1 , . . . . N . These poll rloinials are called Hahn polynomials. Tcl1ebycht.f'~ paper contain-

ing these polynomials was not noticed until long aftcr Hahn rediscovered these polynomials, and found an extension of them.

Hahn was led to these polynornials and an extension of them which contains the polynomial found by Markoff by consideration of the following problem. Hahn 1151 had proven the theorem mentioned earlier about {p,, ( x ) ) and {p:,+, ( x ) ) being orthogonal only for Jacobi, Laguerre and Hermite polynomials. He looked for a similar theorem when the derivative was replaced by the operator

Hahn [16] showed that the general operator in (2.13) could be obtained from its special case when h = 0 . For this case,

Hahn discovered that if { p r , ( r ) ) is a set of orthogonal polynornials and if {Dqp,+, ( x ) ) is also a set of orthogonal polynonlials. then { p , ( x ) ) is one of a few sets of polynomials. Except for a change of scale, the most general of the polynomials is

Here q is fixed, usually real with 1 < q < 1. See Section 3 for a definition of the series in (2 .15) .

The polynoinials Hahn found have other properties in common. They satisf? second order q-difference equations in x which are analogous to the second order differential cquations satisfied by Jacobi. Laguerrr and Hermite polynornials. The

4 RICHARD ASKEY

rnomrmts of a measure of orthogonality satisfy a simple first order equation if the moments are taken with respect to an appropriate set of polynomials. These polynomials may be s'" but in many cases other polynomials need to be used. There is also a Rodrigues type formula. i.e. an analogue of

Hahn only worked out the orthogonality for one of his classes of polynomials. Ortliogonaiity relations have been worked out for all of the polynomials he found, and for a larger class which arise from a more general operator than (2.13). In the next section some of the different methods used to show orthogonality are summarized.

3. Methods of finding orthogonal polynomials and weight functions.

The most general of the orthogonal polynomials with properties similar to those Halm st,udied are balanced 1 9 : ~ basic hyperge~met~ric series.

A basic hypergeometric series is a series

(3.1) CcI,

with CL + 1 /ch a rational function of qh. The most important examplc5 have the form

where

In this case. the series is denoted by

The series (3.4) is balanced when. t = q.

and the series terminates because one value of the n,'s is y " . See 1131 for more on hasic hypergeometric series.

For these balanced 4p:3's. t,here are two cases where thcre is ~rt~hogonality wit,h respect t,o a positive measure. In one case the measure has an absolutely corltirll~ous part. arid may have a finite nunlber of mass points. In the other case. whicli extends the polyiiomials of Chebychev mentioned above, t,he orthogonality is purely discrete. on a finite sct of points.

The second case gives the polynomials

where X(.r) = q ' + qL+'cd and hdq = q ' . ?i = 1 .2 . . . The first case has

q-"3 q ' L - I ' 0

KI(CO~Q) f r 1 4 9 < ab. nr, ad

CONTINUED FRACTIONS AND ORTHOGONAL POLYNOMIALS

where t,he norrnalizat,iori coefficient is giver1 by

(3 .7) !,, = a p " (ab; q),, (ac; q),, ( ad : q) , , . This riorinalization gives symmetry to V,, ( x ) in thc four parameters 0. b. c and

d . As is the case for all sets of orthogonal polynoniials. tliesc polynonlials satisfj

three tern: recurrence rclations. For R,, ( X ( x ) ) this recurrence rclatioil is

(3 .8) [ X ( x ) - X(O)]Rn ( X ( X ) ) = A11 K + I ( X ( J ) ) - (Art + Ci,)Rrt ( X ( X ) ) i- Cit Riz-I ( X ( J ) ) ,

with

( 1 - abqn+ ' ) ( l - nq"+ ' ) ( l - bdqnt ' ) ( l - cqnS I ) All = -

( 1 - nbq2n+l 1 ) ( 1 - abq2n+2 (3.9)

q ( l - q n ) ( l - b q n ) ( c - abqn)(d - n q " ) Cl , =

( 1 - aDq2")(1 - nbq2"+' ) This recurrence relation holds for all J. whcn 7r = 0, 1 . . . . . ,V - 1. arid for .r =

0 , 1 , . . . , N when n = N. where N is given by q = hdq. This is shown in [5 ] . and the orthogonality is then obtained directly from the recurrericc toeffificjents and three other facts. One 1s the symmetry in R,, ( X ( x ) ) 111 71 and .r. If n and .r are interchanged, then ( a , b. r.. d ) is changed to (c . d. a. b ) . Thc second is a. general property of orthogonal pol) nonlials.

If { p , , ( x ) ) is orthogonal with

P,, ( x ) ~ , , , ( x ) M x ) = hllbl,, ,, ,

then the recurrence relation

determines h,, up to a constant by

Finally. a rnatrix which is orthogonal by colunl~is is ortl i~gonal bv rows. See [5] for details of this argunlent .

A second way to find the weight function. or at least guess wlidt lt 15. co111cs froin a different three term relation between orthogonal polynoniials.

Let { p , , ( x ) ) be orthogonal on [a , b] with respect to dn( . r ) . Assuine that r 2 b is finite, and normalize by pi, ( c ) = 1. Then

and

(3.13) q,, ( . ~ . ) q ~ , , ( x ) ( c - . x )da( . r ) = 0 , n~ # 1 1 .

Thc proof is a simple calculation using the orthogonalit\- of {p, , ( . r )} . To see how to use this fact, consider the polynomials V,, (J.) renornlalizcd by

q-" . qrr Inbccl. cle'", ne-'" nh, nc , ntl

where .r = cos 8.

6 RICHARD ASKEY

IVhen aep" = 1, x = ( a + a p ' ) / 2 , p, , ((a + a p ' ) / 2 ) = 1, and (3.15)

qp" ( 1 - q'"abcd)(l - 2ax + a') p,r ( x : a , b, c, d ) - p l t + ~ ( x ; a , b. c. d ) =

( 1 - a b ) ( l - a c ) ( l - a d ) ~ 7 1 ( x ; aq , b. c , d ) .

The three term recurrence relation for p,, ( x ) is equivalent to (3.8) for R,, ( X ( x ) ) . There is a general result which is applicable here. See Chihara and Nevai [ll] for simple proofs. In the present case, this implies that the measure has an absolutely continuous part supported on [ - I , 11 with possibly finitely many mass points off this interval. If there is just an absolutely continuous part. which we call w ( x ; a , 6, c, d ) , t hcn

Iteration gives

with

(3.18)

Since the polj-nomials V,, ( x ) are symmetric in a , b, c , d , the weight function should b r

w ( x : 0 , 0 , 0 ,O) 71!(.r: a , b. c , d ) =

h ( x , a ) h ( x , b ) h ( x , c ) h ( x , d ) '

There are a few special cases which are easy to treat. When a = -b = q , c = d =

q ' / ' , the polynonlials are a constant multiple of sin(n + 1)0 / sin 8. x = cos 0, and the orthogonality of these polynomials is well known. The easiest way to identify this special case is to use the three term recurrence relation. See [6] for details of this argument. The weight function does not have any discrete mass points when -1 < a , b , c , d < 1.

Both of these arguments use three term recurrence relations, but do not use continued fractions. There is one other arguments which should be mentioned here. Its use of a recurrence relation is slightly different.

Let {pA ( x ) ) be a polynomial of degree k in X ( x ) . If d a ( x ) is a positive measure on ( a , 6 ) for which

(3.20)

exists for k = 0 . 1,

satisfies

then

Pl, ( x ) =

. h

CONTINUED FRACTIONS AND ORTHOGONAL POLYNOMIALS

This is obvious, since

'pk(x)p,(x)doc(x) = 0, k = 0,1 , . . . , n - 1

and nz

k=O Szego [[34], Section 2.21 states this in a more general form. He also wrote that

(3.21) was "not suitable in general for derivation of properties of the polynomials in question." This is true, but only if the words "in general" are interpreted cor- rectly. I used to say that this representation was not useful. One of my Ph.D. students, James Wilson, proved me wrong. He showed that for the weight function for either R,(X(x)) or V,(x), it is possible to choose pk(x) SO that the moments can be computed explicitly, and then showed how to take linear combinations of the rows to explicitly evaluate the determinants. Wilson used recurence relations to obtain explicit formulas to evaluate these determinants, and even a more general determinant. The most general one he evaluated gave a new set of biorthogonal rational functions. See [40].

The integral of the measure for p, (x) is a generalization of the beta integral. It is

1 ' h(x, l )h(x , ql/')h(s, - l )h(x, -ql/')dz

h(x, a)h(z, b)h(x, c)h(x, d)d= (3.22)

when -1 < a , b, c, d < 1. The usual notation is

k

(3.23) n ( a , ; d X = ( a l , . . . , ak ;&. 3 = 1

The functions pk(x) used in (3.22) in this case are

4. Continued fractions and orthogonality.

There is an old theorem of Markoff [22] which is useful in finding explicit orthogonality relations when a three term recurrence relation is given. This theorem does not work for all three term recurrences, but it does for many interesting recurrence relations, and gives the values of some integrals 1 do not know how to find in any other way. Exactly how important these integrals and the corresponding orthogonalities are is still not determined, but some seem to be important.

Here is part of Szego's treatment of Markoff's theorem, from [36]. Let A,, > 0, C,, > 0 when n > 1, p-1 (x) = 0 (or C1 = 0) and assume {p, , (x))

satisfies

po (x) a constant.

8 RICHARD ASKEY

Consider the Stieltjes continued fraction

with the same ,4,,. B,,. C,, in (4.1). Let R,, ( x ) / S , , ( x ) he the approximating rational functioris with S,, (1) of degree

n and R,, ( J ) of degree n - 1. Then

(4.3) s,, ( x ) = bpl, (.c)

and

where 3 and 5 are constants depending on the first three moments of da( t )

and & ( x ) is the nleaslire used in the orthogonality relation of p,, (x). hlarkoff's theorem requires that n and b are finite. and that is all that will be needed in the examples below. There is a variant due to Stieltjes which allows b to be infinite when other conditions are added.

hlarkoff proved that

RO ( x ) = p ( x ) lim --- "-" S,, ( S )

when s is in the complex plaric cut on the interval n < x < b. The function E'(x) is dcfined hj-

Smgo's proof of (4.6) uses the theorcrn of Stieltjes that Gaussian quadrature couvergcs to the integral of the fiinction when the functioii is continuous. See [36] for detail\.

The first non-classicnl example I know was done by Sherman. and involves associated Legtndrc fiinctions.

Legcndrr pol? rloriiials 5atisfy the three term recurrence relation

The associated Legelidre polvnornials arise whrn all the n's in the coefficients in this recurrence relation are shifted by the same arriount. Sherinan [33] treated the most important case, wlien tllc shift is by oiie. Thc general case is as follows.

P_1(x;c) = 0, f i ( x : c ) = 1. IVhrn c = 1 . Sherman showed that

(4.10) P,, ( I . : l )Pr, , (.r; 1) [ ' 7r + [ log ( l + ' r ) ] ' ] ' d . r = ~ . -- r n f n . 1 - s

Shcrinan did rriort, grneral work. trcating some associatrd Jacohi polynomial when the shift is bj- 1. Sccl Nevai [25] for gc~nr~ral results with integer shifts.


The general case was treated twice. Once by Pollaczek [28] as part of a more general set of orthogonal polynomials wliich will be inentiorled next. Tlw otlwr time was by Barrucand and Dickerison [8]. The form of tlw weight fun( tion given in these two cases is different. Barrucand and Dickimon have

as the weight function. The function PC (.r) and Q, (.r) are Legendrc~ fiinction5 of the first and second kind. Thr polynornials when c = 112 and c = 312 arose In finding polynomials orthogonal with respect to the ineasures

(1 - y y l L ( 1 - k.2 .L -1IL J 1

111 work done by Revs [30]. In a remarkablr seric.s of short notes [27]. [28], and others. and a monograph

[29]. Felix Pollaczck described a vorj interestmg 5ct of orthogonal polynomials which he discovered. One set satisfies the following recurrence relation.

P - ] ( r ) = O . P,,(r) = 1. Inltlally he found these pol\ nomi& u hell = f d i d r = 0 G a h r %ego

[35] appreciated the import ante of t he111 and cxtcndrd-~o1l;iczrk's rcwdt to iiic lude A. Then Pollaczek added the c. Thtl assoclatrd Legciltlre pol\ nolnldls arise mlicn a = b = O a r l d X = L.

Il'hen n > lbl. 5~ + c > 0 nncl ( > 0 or whcn n 2 lbl. 2X + c > 1. c > -1. the orthogonality rtllatioll is 11 it11 respwt to an dbsolutclj continuous rricasurc given for l < J < l b y

wit 11

(4.14)

Thc case c = O is a littlc easicr to handle. and contains the lnost iinportant c~xainple. The weight function satisfic.~

r ]

\I'hcn the integral in (3.15) is finitc. u3(s) # O for -1 < .x. < 1. ant1 u , ( r ) is smoot,h enough. Szcg6 has shown that the polynonliitls orthogonal with rvspect to t,his weight function look a lot like .lacobi polynoinials. SIT [34]. Tlirse polylmrnials whrn taken to be orthonormal grow at most like a poxwr of 11 whcn .r. = 1. I'ollaczck polyrlomials havc more rapid growth when either n # 0 or 0 # 0. Thesc werc the first polynoiriials to b r fouiid explicitly m-hose weight function is not in the Szegii class.

Szegij gave the problcm of finding out mar(' about Polla,czek's polynomials to a Ph.D. student. A. Novikoff. Novikoff's thrsis [26] was not p~lblished. but Szegii included a sumniary of the mait1 results in an appeidix t,o t,lie second edit ioii of [34]. Essentiall>- nothing was tlo~io with t,liesc, polynomials for many y ~ a r s .

Thc first rcxlated work was dolic, by Srlliah [32]. this was not plhlislled. and the connect,ion with I'ollaczek's work was not rcdizcd. St,lliali was cxt.ending ~vork

10 RICHARD ASKEY

of Karlin and 1LlcGregor. [19], who had come across a limiting case of Pollaczek's polynomials but with different conditions on the parameters. The Karlin and Mc- Gregor work dealt with a set of orthogonal polynomials whose weight function is supported on a discrete set of points with 0 as the only limit point. See [19]. Selliah found the Stieltjes transform of the measure of orthogonality, and from this found all of the discrete mass points. However, his lack of knowledge of Pollaczek's work led him to write: "It seems very unlikely that this [the Stieltjes transform] will in general give us on inversion, a simple function for the distribution."

Askey and Isinail knew of Pollaczek's work, but not that of Selliah. when they studied the special case of (4.12) when b = c = 0. Here is a sample of the results in 141. A different normalization was used to make it easier to consider the limiting . >

case when the measure is purely discrete. Define the polynomials G I , ( x ) = G,, ( x : a , b) by

G 1 ( 2 ) = 0. G o ( x ) = 1. There is orthogonality with respect to a positive measure when a + b > 0 and

a > O , o r w h e n a < - l a n d a + b + l < O . L$'hen a > 1. b > 0, the orthogonality relation is contained in the ones Pollaczek

[27] and Szego [35] found. When 0 < a < 1, b > 0, in addition to an absolutely continuous part of the measure used for orthogonality, there are infinitely many discrete mass points outside the interval [-d, dl, d = 2&/(a + 1) . which have d and -d as limit points.

To understand the power of Markoff's theorem combined with other 19~' ' century parts of analysis, try to evaluate the integral and sum in the following orthogonality relation. I do not know how to do this directly, or know if the integral and sum can be evaluated separately.

THEOREM 4.1. If 0 < n < 1, b > 0 and G , , ( x ; a , b) = G , , ( x ) 2s defined by the recurrence relntron (4.161, then

where

b b ( l - a ) r ( % +i- 2a

cot 8 ) 1 2 , and


After this paper was written, a number of other examples were worked out. The parameter b in (4.12) was reintroduced.

A different type of generalization of Legendre polynomials had been found. first by L. J. Rogers [31], who was unaware he had orthogonal polynomials, but who found many other remarkable formulas for his polynomials. His polynomials, which are now called continuous q-ultraspherical polynomials. satisfy the recurrence relation (4.20)

2 71-1 2 4 1 - Pq'"C,,(x:P I 9) = (1 - qn+')C,+1(x:P I 9 ) + (1 - P q ) G - l ( x ; P I 91,

C-l(.GP 19) = 0, Co(x;P 19) = 1. Those polynomials as orthogonal polynomials were rediscovered by Feldheim

[12] and Lanzewizky [20]. They were also rediscovered by Allaway [3]. Allaway was more careful in giving all of the cases when orthogonal polynomials arise, and found a new set of polynomials which are a limiting case of those of Rogers. There are two limiting cases, one of which is found as follows.

Let WA be a kt" root of unity, say WL = exp(2nilk). Set P = s " ' + ' u J ~ , q = SUL

in (4.20). Divide by 1 - sw;" and let s -+ 1. The resulting polynomials B;(x; k) satisfy

B$(x;k) = 1. BF(x;k) = 22 if k > 2. See [2] for the orthogonality of these polynomials. Polynomials whose recurrence relation coefficients are constant except on an

arithmetic progression, are frequently called sieved orthogonal polynomials. Both q-extensions and sieved extensions of the results in Theorem 4.1 are con-

tained in the following papers, [I], [9]. Also see [17] for related results. Another paper which builds on work of Pollaczek is [7]. Here, associated

Laguerre and Hermite polynomials are treated as limits of other polynomials of Pollaczek, his extensions to associated polynomials of polynomials first found by Meixner. For associated Hermite polynomials defined by

2xHn (x; c) = H,,+I (2; c) + 2(n + c)Hn-, (x: c), (4.22)

H-, (2: c) = 0, Ho(x: c) = 1,

t,he orthogonality is

with

When c > 0 the weight function can be written as

12 RICHARD ASKEY

Various Pad4 type approximations havc been found using these poly~iomials. and int,eresting Stieltjes transforms follow from t hcse results. See [7]. [41], [42] for examples.

Fillall\: some itnpressive contirmcd fractioris stated bj- Ranianujari and first proved and extended bj- IVatson werc reconsiderrd by Alasson [23], [24]. He was able to extend t,he most general t,o an infinite c~nt~ilrned fraction. while Ramanujan and II;;rtson old\- gave their rcsult,s for ternlinating continued fract,ions. hlassori also found a coinpanion continued fract,ion which had been rnissed. If one likes beautiful fornullas, and t,hcse continutd fractions are beautiful. then it is useful to have a firm knowledge of continued fractions.

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731 737, reprinted in Gabor Szego, Collected Papers, volume 3, 225231, Birkhauser, Boston, 1982.

[36] G. Szego, A outlzne of t he hzs tory of orthogonal polynomzals, in Orthogonal Expansions and Their Continuous Analogues, D. T. Haimo ctiitor, Southern Ill. U. Press, Edwardsville, 1968, 3 11, reprinted in Gabor Szego. Collected Papers, volume 3, 857-865, Birkhauscr, Boston. 1982.

[37] P. L. Tchebychef, S u r les f m c t 7 o n ~ contznues , J . Math. Pures Appl. series 11. 3 (1858), 289 323. Translation of Russian paper published in 1855; reprinted in Oeuvres de P. L. Tchrbychcf. tome I. eds. A. hlarkoff and N. Sonin, St. Petersburg, 1899; reprint,ed Chelsea, N. Y. . 1961. 201-230.

[38] P. L. Tchebychef, SILT L 'znterpolatzon des valeurs , equzdzstantes. in Oeuvres de P. L. Tcheby- chef, Tome 11. etls. A. hlarkoff and N. Sonin, St. Petersburg, 1899; reprinted Chelsea. N.Y.. 1861, 219 2 U .

[39] X. G. Viennot, U n e theorze combinatozre des po lynomes or thogon~uz~r yenerauz , Ut~iv. of Quebec, hlontreal. 1984.

[40] J. A. Wilson, Orthogonal functzons f r o m G r a m de te rminan t s . SIAhl .J. hlath. Anal. 22 (1991). 1147 1155.

[41] J. Wimp, S o m e explzcit Pad6 approx iman t s f o r t he functzon @'/a and a related quadraturr f o rmula znvolvzng Bessel fun,ctions, SIAM J . hlath. Anal. 16 (1985), 887 895.

[ a ] J. Wimp, Explp2lczt fonnu1a.s for t he assoczated Jacob7 polynomzals and s o m e applzcatzons. Can. J. Math. 39 (1987). 983 1000.

DEPAWRIEST OF ~ I X ~ I I E ~ I A T I C S . UNI\.SRSITY OF \VIS?ONSIN ~ I . \ I ) I S O N . 1 I ~ 1 1 1 ~ 0 4 . \VI 53706 E-mazl address: askeymmath. wisc . edu

Contemporary M a t h e m a t ~ c s Volurrw 236. IWS

The Problems Submitted by Ramanujan to the Journal of the Indian Mathematical Society

Bruce C. Berndt , Youn Seo Choi. and Soon--Yi Kang

To .Jerry Lange on his 70th bzrthday

Abstract

Between 1911 and 1919 Ramanujan submitted a total of 58 problems to the Journal of thr Indzan Math~matzca l Soczety. hlany of these problems have become famous since their inclusion with Rarnanujan's Collected Papers published in 1927. In this paper we give a survey of all 58 problems, emphasizing the mathematical activity generated by them since 1927.

1. Introduction

Between the years 1911 and 1919 Ramanujan submitted a total of 58 problems, several with multiple parts, to the Journal of the Indian Mathematical Society. For t,he first five. the spelling R a m a n u j a m was used. Several of the problems arc elementary and can be &tacked with a background of only high school mathematics. Ebr others, significant amounts of hard analysis are necessary to effect solutioiis. and a few problems have riot been completely solved. Every problerri is either interesting or curious in some way. All 58 problems can be found in Rarnarnljan's Collected Papers [172, pp. 322-3341, As is customary in problems sections of journals, edit,ors norrrlally prefer to publish solut,ions other than those given by t,he proposer. However, if no one other t,llan the proposer has solved the problem, or if the proposer's so1ut)ion is ~art~icularly elegant,. then the proposer's solution is published. This was likely the practice followed by the editors of the Journal of the Indian Mathernatical Societ?y, but naturally thp editors of Rarnanujan's Collected Papers chose different criteria; only those printed solutions by Ramanujan were reproduced in his Collected Papers.

The publication of the Collected Papers in 1927 brought Ramanujan's problems to a wider matherriatical audience. Several problems have become quite farnous and have attracted the attention of many mathematicians. Some problems have spawned a p1et)hora of papers, nlany cont,ainirig generalizations or analogues. It

1991 Mathematics Subject Classzjication. Primary 11-03, 40-03; Secondary llD25, l lD99,

40A15.

@ 199'3 American Mrrthematlcal Society

15

16 BRUCE C . BERNDT, YOUN-SEO CHOI, AND SOON-YI KANG

thus vems appropriate to provide a survey of all 58 problems irldicating the activity gt~lerat td b) t11v prohlcms sirlcr 1927.

In referring to these problcms, we follow the numbering given in the .Journal of the l n d ~ a n Matherrmtzcal S o c ~ r t y . Although the division of problerns into categories is dlma~s somtwhat arbitrary, we have decided to place the 58 problems in nine subsets as follows:

Solutions of Equations: 283. 284. 507, 666. 722 Radicals: 289. 524. 525, 682, 1070, 1076 Further Elementary Problems: 359. 785 Number Theory: 427, 441, 464, 469. 489, 584, G29. 661, 681, 699. 770, 723, 784 Integrals: 295. 308. 353, 386. 463. 739. 783 Series: 260. 327. 358. 387, 546. 606, 642, 700. 724, 768. 769 Continued Fractions: 352. 541. 1049 Other Analysis: 261. 233, 526. 571, 605, 738. 740. 753. 754 Geometry: 662. 755

Some of Ramam?jan's problems have been slightly rephrased by the editors of his Collected l ' q m - s . Cherally, wc cluot>e either Rarnanujan's forrnulatiori of each problem. or th t version in the Collected Papers. However, we have t,aken the lib- crtl- of replacing occasional archaic spelling by more contemporary spelling. and nmst often w t have crnployed surrlrnation notation in place of the more elaborate notation n l + nn + . . . . After the number of the question, t,he volume and page nuniber(s) where the problem first appeared in the Journal of the Ind ian Mathe- rrlntical Society. which we abbreviate by JIhIS, are stated, and t,hese arc followed by the volurnc(s) and page number(s) where solutions, partial solutions, or cornrnents are givtm \VP do not cite problems individually in t,he references of this paper. \ire also do not separately list in our references the solvers of the problerns cited in Hamarilijan's Collected Papers [172]. However, if a solution was pliblished after the publication of the Collected Papers in 1927, then we record it as a separate item in the bibliography. AIany of the problenls. or portions thereof, can be found in Ramanu,jan's not,cbooks [171]. Normally in such a case. we cit,e where a problem can be located in the notebooks and where it can also be found in Berndt's accounts of t,lw notebooks [20]-1241.

2. Solutions of Equations

QI-ESTION 28:3 (.JIRIS 3. P. 89: 3, PP. 198-200; 4, P. 106). Show that it is possible t o solve the rquatsons

nihere a,, g , z . p. q . r ar.e the unknowns . Solve the above when n = 2, b = 3, c = 4, d =

6. c = 12, wnd f = 32.

T H E PROBLEMS SUBMITTED BY RAMANUJAN

Question 283 is a special case of the more general system

where X I , x ~ , . . . , x,, and yl. y 2 , . . . , y, are 2n unknowns. ingeniously solved by Ra- manujan in his third published paper [164], [172, pp. 18-19]. Implicit assumptions were made in Ramanujan's solution, and thus it should be emphasized that (2.1) is not always solvable. For a sketch of Ramanujan's solution. see a paper by Berndt and S. Bhargava [25]. Another derivation of the general solution for (2.1) was found by hI. T. Naraniengar [148]. The more general system (2.1) is also found on page 338 in Rainanujan's second notebook [171]. We quote the discussion of (2.1) from Berndt's book 123, p. 301.

"It is easy to see that the system (2.1) is equivalent to the single equation

Thus. hmanujan ' s problem is equivalent to the question: When can a binary (2n - 1)-ic form be represented as a sum of n (2n - 1)th powers? In 1851, J. J. Sylvester [196], [197], [198, pp. 203-216, 265-2831 found the following necessary and sufficient conditions for a solution: The system of n equations,

must have a solution ul , uz, . . . . u,,+l such that the n-ic form

can be represented as a product of n distinct linear forms. Thus, the numbers X I , x l , . . . , xI i . y1, y 2 , . . . , yn are related to the factorization of p(w, 2 ) . Sylvester's theorem belongs to the subject of invariant theory, which was developed in the late nineteenth and early twentieth centuries. For a contemporary account. but with classical language. see a paper by J. P. S. Kung and G.-C. Rota [113]."

QUESTION 284 (JILTS 3. P. 89: 4, P. 183). Solve

This problem is the special case a = 6, b = 9 of the more general system

18 BRUCE C. BERNDT, YOUN-SEO CHOI, AND SOON-YI KANG

recorded by Ramanujan on page 338 in his second notebook and solved by Berndt in [23, pp. 27-29]. The solutions comprise 25 values for both x and y. As pointed out in [23, pp. 28-29]. Ramariujaa's published solution. which is reproduced in [172, pp. 322-3231. contains some mistakes. \Ve very briefly describe the solutions. Let

s = o + f i + ? and , y = a / ? + p y + y a ,

where a i j y = 1. Then [23, pp. 28-29] a 5 , [ji3', and y' are roots of the equation

t J - a t 2 + bt - 1 = 0.

Ramanujan then listed the values of s as

a+P+r , ~ + P P + ? P ' . a + P p L + y p J . ~ + P P + ? P ' .

cup + Pp" +p2\ ap2 + pp4 + ? p i , ap ' + Pp' + ?P4.

whcre p is a primitive fifth root of unity. However, observe that the third member of this set may be derived from the second member by replacing p by p2. Also. the seventh can be obtained from the fourth by replacing p by p'. Lastly, the sixth arises from the fifth when p is replaced by p2. Ramanujan missed the values

a p + P + 7 p 4 , (YP + Pp' + 7, o p + Pp5 + ~ p .

QUESTION 507 (JIMS 5, P. 240; 6, PP. 74-77). Solve completely

and hence show thnt

I11 parts (a)-(c), the signs under the outward-most radical sign have period three; they are, respectively, -, +, -; -, +, -; -, +. +.

Rarnamijan's published solution in the .Journal of the Indian Mathematical Society is correct,. However, there are four sign errors in the solution published in his Collected Papers [172, pp. 327-3291, which we now relate. The equations in (2.2) easily imply that .r ~at~isfies a polynomial equation of degree 8. This polynomial factors over ( ) ( d m ) int,o a quadratic polynomial c2 - T - u and t,wo cubic polynoinials, which are given in the Collected Papers near the end of Rarnanujan's solution on page 328 and just before the verifications of the three exairlples (a)- (c). Each of t,hese two cubic: polynomials has two sign errors. The polynomials are corrr:ctly given in Ramanujan's original solut,ion and in Berndt, '~ book [23, pp. 10, 11, eqs. (4.2), (4.3)].

The formulation of Question 507 indicates t,hat the three examples can be deduced from the solutiorls of (2.2). However, in his published solution, Ramanujan did not do this but established each ident,it,y ad hoc. In his so1ut)ion to Question 507. M. B. Rao [176] derived each of the examples, (a)-(c), from t,he general solut,ion of (2.2).

THE PROBLEMS SUBMITTED BY RAMANUJAN 19

On pages 305-307 in his second notebook [171]. Rarnanujm offtm a more extensive version of Question 507 [23, pp. 10-201. By taking successive square roots in (2.2). it is not difficult to see that we can approximate the roots by ail infinite sequence of nested radicals

However. we must be careful in t,aking these square roots, for there are three different square roots to be taken and t,here are two choices for the sign of the square root in each case. The 2" different sequences of nested radicals correspond to the eight root,s of thc octic polynomial arising from (2.2). Of course, one must determine the values of a for which the eight infinite sequences convergc. In [23, pp. 14-16]. it was shown that the infinite sequences converge for a > 2. which is not tlle best possible result. For example, it was indicated in [23, p. 151 t,hat the sequence

wit,ll signs +, -,+ of period 3, likely converges at least for a > 1.9408. In our discussion of Quest,iorl 289. we will return to the question of the convergence of infinite sequences of nested radicals. .

For a part,icular value of a , one can numerically check which infinite sequence of nested radicals corresponds to a given root. In general, for the t,wo infinite sequences of nested radicals arising from the two roots of the quadratic polynomial. the identification is easy. However. for the remaining six roots, the problem is more difficult. On pages 305-306 in his second not,ebook. Rarnanujan made thcse general identifications [23, p. 17, Entry 51. In [23, p. 181, asymptotic expansions of the six roots and the six sequences of nested radicals. as a + x, were established in order to obtain the desired rnatchings.

There is also a very brief discussion of the system (2.2) in I.'. Cajori's book [53, pp. 196-1971,

QUESTION 722 (JIILIS 8 , P. 240). Solve completely

and deduce that, z:f

then

and that, 2f


then

Question 722 is obviously an analogue of Question 507. The first complete solution to this problem was given by XI. B. Rao [176] in 1925. He also solved Question 507. examined the analogue for five equations of the same type as Questions 507 and 722, and presented several examples. The second solution was given in 1929 by G. N . Watson [210]. who also derived Ramanujan's two examples. A considerably shorter solution was found by A. Salam [I831 in 1943. Question 722 can also be found in Ramanujan's third notebook [171, vol. 2, p. 3671, and a fourth solution is given in Berndt's book [23, pp. 42-47, Entry 32, Corollary].

QUESTION 666 (,JIILIS 7 , P . 120; 8 , P. 31 ) . Solve i n posztive rational numbers

3 For ezample, x = 4 , y = 2 : ~ = 3 ~ , y = 2 : .

Since the published solution by J. C. Swaminarayan arid R. Vythynathaswarny is short and elegant and was not published in Rarnanujan's Collected Papers, we reproduce it here.

Put x = k y . It follows that y h p l = k .

I t is easy to see that k is a rational solution if and only if k = 1 + l / n for some positive integer 1 1 . Thus.

%'hen 71 = 1 . x = 4 and y = 2: when n = 2 , z = arid y = 2. 3. Radicals

QUESTION 289 (JIILIS 3 , P. 90: 4 , P. 226) . find the value of

(ii)

The values of (i) and (ii) are 3 and 4, respectively. Rarnanujan's solutions in volume 4 [172, p. 3231 are not completely rigorous. A note by T. Vijajaraghavan at the end of Appendix 1 in Rarnanujan's Collected Papers [172, p. 3481 justifies Rarnanujan's formal procedure. This note was considerably amplified in a letter from Vijayaraghavan to B. hf. \$'ilson on January 4. 1928 [36, pp. 275-2781. If a , > 0 , l < J < x, then Vijayaraghavan proved that a sufficient condition for the convergence of the sequence

THE PROBLEMS SUBMITTED BY RAMANUJAN

is that - loga,, lirn - < X.

r r - x 2"

Vijayaraghavan's criterion for the convergence of infinite ~iested radicals is a problern in P61~-a and Szegij's book [156, Prob. 162, pp. 37, 2141. A. Herschfeld [96] also proved Vijayaragliavan's criterion. In a 1at)er question sub in i t td t,o the Jour- nal of the Indian Mathenmtzcul Society, Vijayaragllavan [204] claimed a stronger theorem for t,lle convergence of t,, , which he said was best possible. However, the problenl is flawed, and no correction or solution was evident,ly publishrd. However. Problem 163 in P61ya and Szego's book [156, pp. 37, 2141 is likely thc criterion that Vijayaraghavan had in mind. A sufficient condition for c:onvergencc wherl thtl sequence {a, ) is complex has been established by G. Schuskr arid \V. J. Tllron [189]. Convergence criteria for certain iiifiriit,e nested radicals of pth roots liaw been given by J . LI. Borwein and G. dc Barra [42].

Part (i) appeared as a problem in the \Villiam Lowell Putiianl coinpt'tition in 1966 [141]. The values of (i) and (ii) appear as examples for a gcncral tlicwrcm of Rarrianujan on ncsted radicals in Section 4 of Chapter 12 in his second notebook [21, p. 108, Entry 41. Further references for Qurstioii 289 and relatcd work arc found in Section 4, and addit,ional r,xamples of infinite ~ ics t td radicals appear in Entries 5 and 6 in that same chapter [21, pp. 109-1121.

QUESTION 523 (JIILIS Ci. P. 39: 6. PP. 190-191). Show thof

Part (i) appears on the last page of Rarnanujan's second notebook [171, p. 3561. The proofs of (i) and dnother co~npanion result by Berndt 111 [23, p. 391 use a general result of Rarnanujan [23, p. 22, Entry 101 on the sunl of t h ~ cube moth of the three roots of d L U ~ K polyno~nial. The proofs of (i) and (ii) b\ N. Salikara Aiyar in volume 6 of the Journal of the I n d ~ a n Mafh~rnntzcal So( rely arc. similar

QUESTION 682 (JIhIS 7, P. 160; 10. P. 325). Show how to find t h ~ u ~ b e root.^

of surd.^ of the ,form A + m, and d ~ d u c e that

Qr EsIION 525 (JIhfS 6. P. 39: 6, PP. 191-192) Show h o ~ to find t h p 5quclr t

roots of surds of the form a + m, and h r n w prow that


QUESTION 1070 (.JIhIS 9, P. 160; 16, PP. 122-123). Show tha t

QUESTION 1076 (JIhfS 11, P. 199). Show tha t

6)

(ii)

Of course, one can establish each of the nine identities in the four preceding problems by taklng the appropriate powcr of each side of each equality. applying the rnultinornial thcorcin, and simplifying. However, such a proof provides no insight whatsoever into such an equality. nor does it indicate how Ramanujan might have discovered it. Both the left and right sides of each of the equalities are units in sornt. algebraic number field. Although Ramanujan never used the term unzt, and probably did not formally know what a unit was, he evidently rralized their funcldinental properties. He then recognized that taking certain powers of units often led to elegant identities.

Berndt, Chan. and Zharig [33] have found generalizations of the identities above. For example. for any real number a ,

and

(a + 2) + (1 - 4a) \i/2aL-*-l 9 3

which when we set a = 5 in the former equality and a = 7 in the latter yield (i) and (ii), respectively. of Question 525. As another example, for any real number a ,

which wit,h a = 2 yields t,he equality in Question 682. In both the original formulation and the Collected Papers [172, p. 3341, the

exponent,^ 116 and 118 on the left sides of (i) and (ii) in Question 1076 were unfortunately inverted. The inversion likcly was not irnrnediat,ely discovered, as the first solution t,o the corrected problems was not given until 1927 by S. Srinivasan [194]. In 1929. a second, shorter solution was given by R. Kothantlararnan [110].


On page 341 in his second notebook [171], Ramanujan stated two general radical identities involving cube roots. For example,

See [23, pp. 34-36] for these two identities and several additional examples of the sorts we have discussed here.

Many papers have been written on simplifying radicals. In particular. we mention papers by R. Zippel [224] and T. J. Osler [151]. S. Landau [116]. [117] has given Galois explanations for several types of radical identities.

4. Further Elementary Problems

QUESTION 785 ( J IMS 8. PP. 159-160; 8, P. 232): Shour that

This i s analogous to

(4.2) (2 {(a' + b2)'/' - a} {(a' + b ) - b})li2 = a + b - (a2 + b 2 ) l l 2

The proof by K. K. Ranganatha Aiyar, R. D. Karve, G. A. Kamtekar, L. N. Datta, and L. N. Subramanyam is clever and short, and so we give it.

In the identity

(a + b - r)" = ( a + b)" - rL 3r (a + b)' + 3r2 (a + b) ,

put r" = a" + b:'. Thus,

(a + b - r)" =3ab(a + b) - 3r(a + b)' + 3r2(a + b)

=3(a + b)(r - a ) ( r - b).

Hence,

from which (4.1) follows. Equality (4.2) can be proved in a similar fashion.

QUESTION 359 (JIMS 4, P. 78; 15. PP. 114-117). If

prove that

(+ sinrcosi)" '+ f+roszs inz)" ' = (sin2t~)'/",

and verzfy the result when

sin 2x = ( & - 2 ) " ( 4 + ~ % ) ~ , sin 2y = &-2. sin 22 = ( & - 2 ) " ( 4 - 6 ) 2 ,

Note that it took over ten years before a solution was submitted. Nore recently, another solution was given by V. R. Thiruvenkatachar and K. Venkatachaliengar


1203, pp. 2-91. but their solution is also quite lengthy. It would seem desirable to have a briefer, more elegant solution, but perhaps this is not possible.

5. Number Theory

QI;ESTION 441 (JIILIS 5, P. 39; 6, PP. 226-227). Show that

and find other quadratic expressions satisfying similar relations.

This problem gives a two-parameter family of solutions to Euler's diophantine ccluation

The published solut~on hy S. Narayanan, in fact, gives a more general family of solutions; if 0=X(X3+1) , rn = 2 X ' - l , n =X(X3-2 ) , a n d p = X 3 + l , then

Equation (5.1) is obtained by setting X = 2 above. The equality (5.1) is mentioned by Hardy and Wright 193, p. 2011. C. Hooley

1991 employed (5.1) t,o obtain a lower bound for the number of integers less than z that can be represented as a sun1 of two cubes.

Quest,ion 441 can also be found on page 266 in Ramanujan's second n~t~ebook [23, p. 561. In his second n~t~ebook. Ramanujan gave further families of solutions to (5.2). First, he gave t,wo families of solut,iorls which include (5.1) as special cases 123, p. 54, Entry 1; p. 55, Entry 31. Second, Rarnanujan gavt, another t,wo- pararn~t~er family of solut,ions to (5.2) and several examples [22, pp. 197-199, Entry 20(iii)].

According to Dickson 173, p. 5501, the problem of finding rational or integral solutions t,o (5.2) can be t,raced back t,o Diophantus 1741. Euler [77]. 179, pp. 428-4581 found the most general family of rational solutions to (5.2). Rarnanlijan, in his t,hird notebook 1171, vol. 2, p. 3871, 123, pp. 107, 1081 also gave t,he most general solut,ion of (5.2). but in a different formulation. Both Hardy [92, p. 111 and Watson [213, p. 1451 were unaware of this entry in the notebooks and so did not realize that Rarnanujan had found t,he most general solution of (5.2). There arc. in fact, several forrns of the general solution: Hard>- and Wright 193, p. 2001 prescnt one due to A. Hurwitz. Further references to gerlcral solutions of (5.2) can be fourid in Dickson's Hzstory [73, pp. 550-5611,

The complet,e characterization of all integral solutioris t,o (5.2) is an open problem. However, C. Slindor [186] has solved the problem if one restricts t,ht solutions to be r~present~ed by quadrat,ic forms in 2 variables: note that Ramanujan's solutions in Question 441 arc represent,ed by quadratic: forrns. Sdndor's paper also cont,ains a summary of further (especially recent) progress on the problem of finding all integral solutions t,o (5.2).

QUESTION 661 (JIMS 7 , r. 119; 13. PP. 15- 17; 14, PP. 73-77). S o l w i n integers


Obviously. (5 .3) is a special case of Eulcr's diophantine equation (5 .2) . The solutions in both volumes 13 and 14 of the Journal of the Indian Mathr-

nmtacal Soczefy are by S. B. hlltra. In volume 13. hlitra presented four nlethocls for obtaining solutioris of (5 .3 ) . The first family of solutions coiltains the special case 8 ' + 6' + 1' = 3'. The second family contains as spcclal cases all examples in the first column above, and the second example in the second column. In volurne 14. Alitra established the general rational solution of (5.3) . Howevcr. the most general solution in m t e g e ~ s is not known.

QUESTION 6 8 1 (JIMS 7 , P. 160: 1 3 , P. 17: 14 . P P . 73-77) . Solve In tnfegers

and deduce th,e following:

Clearly, (5.4) is a particular instance of Eulcr's diophantine equation (5.2) . Observe t,ha.t the second example given above gives t,he famous %xi-cab"

representations of 1729 [172, p. xxxv]. [22, p. 1991. As in the previous problem, the solut,ions in both volumes 13 and 14 of the

Journal of the Indian Mathematical Society are by S. B. Llit,ra. In volurne 13. hlitra established a one-yaramcter family of solut,ions yielding the first two exan- ples abovt.. In volume 14. he derived the general rational solution of (5 .4 ) . h1. Venkata R a n a Ayl-ar [15] found other met hods for examining Quest ions 441. 66 1. and 681. In an earlier paper [16]. he and LI. B. Kao foimd further solntioils to (5 .4 ) . Some very special solut,ions to (5 .4) are found in Ramanujau's lost notebook [173, p. 3411. hl. Hirschhorn [97]. [98] has supplied a plausible arguilient for how Ranlariujan might have discovered these particular solutions.

In contrast to Question 661, Qucst,ion 681 has at,tractcd corisiderable att,mtioii in t,he literature. R.. D. Carrnichael [55] raised the problem of finding t,he general integral solution of ( 5 . 4 ) , and H. C . Bradley [44] asked if t,he om--paramctcr family

of solutions (also found h? hlitrn) constitutes all integral solutions to (5 .4 ) . They further~riore asked for a general solution, if ( 5 5 ) does not give all integral N. B. hlitra [I421 negativel) n~iswcrcd the first question by constructing another famil) of solutions. Unawarc of the work of Ranlari~ljan. Carrnicliael, Bradley, and hlitra. in a letter to K. hIahler, L. J. hlorclell asked if (5.4) had other solutions besides the trivial ones r = 1, y = -a. In response, hlahler [136] con5tr1lcted the family of solutions (5 .5 ) . D. H. Lchmer [128] showed how to construct infinite sequmces of solutions to (5 .4) . In part iculnr, starting from thr paramc>t ric solution (5 .5) . he derived an infinite sequence of parametric solutions. the si~nplcst being

= 2 1 3 ' r . 1 " 2 i 3 1 r i - 3 1 r '+3r. = 2 L 3 ' r ' ) + ~ ' 3 i r 0 3 L r ' + l . = 2 1 3 i r 1 0 - 3 ' 5 r L ,


Supplenienting the work of Lehmer, H. J . Godwin [86] and V. D. Podsypariin [155] found further families of solutions. Finding a complete description of all zntegral solutions to (5.4) appears to be an uri~olved problem.

It 1s natural to generalize Question 681 by asking which positive integers n are the sum of three cubes. For early references, consult Dickson's Hzstory [73, pp. 726-7271, Many papers have been written on this question. and we cite only a few of them. It is conjectured that, if C is the set of all integers representable as a sum of three cubes. then C has positive density. The best result to date is b> R. C. Vaughan [202] who showed that

for each F > 0. Extensive computer searches for solutions of

for various n, have been conducted by several mathematicians. including D. R. Heath-Brown. ItT. hI. Lioen. and H. J. J . te Riele [95] and A. Breinrler [45]. Up to the present time. the most extensive calculations have been performed by K. Koyama. Y. Tsuruoka, and H. Sekigawa [ I l l ] .

QUESI'ION 464 (JI1LLS 5 , P. 120: 5, PP. 227-228). 2" - 7 7s a perfect squore for the values 3,4 ,5 .7 , 15 of n. Fznd other values.

The equation

is called Ramanujan's diophantine equation, or the Ramanujan-Nagell diophant~ine equation, and is perhaps the most famous of the 58 problems that Rarnanujari submitted to t,he .Journal of the Indian Mathematical Society. It should be emphasized that t,he bsol~ition" by K. J . Sanjana and T. P. Trivedi in volume 5 offers a systern- atic derivation of the five given solutions but does not show that these are the only solutions (as the authors make clear).

Unaware of Rarnanujan's problem. IV. Ljunggren [132] proposed the same problem in 1943. T . Nagell's name is att,ached to (5.6) because in 1948 he solved Ljunggren's problem and therefore was the first to prove Ramanu,jan's "conjecture" that no ot,her solutions exist. However, since his paper was writt,en in Norwegian in a relatively obscure Norwegian journal [145]. very few ~nat,henlat,icians were aware of his solution. Thus. after Th. Skolem, S. D. Chowla, and D. J. Lewis [193] used Skolem's p-adic nlethod to prove Ramanujan's conjecture in 1959, Nagell repub- lished his proof in English in a more prominent journal [146] and pointed out that he had solved the problem much earlier than had Skolem. Chowla, and Lewis. Also. in 1959, unaware of Nagell's work, H. S. Shapiro and D. L. Slotnik [192], in t,heir work on error correcting codes. found all solutions of an equation easily seen to be equivalent to (5.6). Significantly improving the methods in [193], Chowla, hI. Dunton, and Lewis [66] gave in 1960 another proof of Ranxmujan's conjecture. In 1962, using the arithmetic of certain cubic fields, L. J . blordell [143] gave another. less elementary proof. hl. F. Hossain [100]: W. Johnson [105]. G. Turnwald [201], and P. Buiidschuh [52] are some of the aut,hors who have also found proofs. In his book [144]. hlordell gave Hasse's simplification [94] of Nagell's proof. Instruct,ive surveys of all known proofs with a p1et)hoi-a of references have been writt,eri by E. Coheii [68] and A. hI. S. Rarnasamy [175].

T H E PROBLEMS SUBMITTED BY RAMANUJAN 27

hlany generalizatioris of (5.6) are found in the literature. We confine further remarks to the generalued Rnmnnujan-Nagell equation

J. Browkiri and A. Schinzel [49] proved that 2" - D = y2 has at most one solut,ion if D $ 0,4 .7 (mod 8). Furthermore: if any solut,ion exist,s. then n < 2. They also pointed out that, in fact, in 1956, they [48] had completely solved a diophantine equation, easily shown to be equivalent t,o (5.6). and so had given the second solution to Rarnanlljan's problem. R. Apiiry [8] showed t,hat. if p is an odd prirne not dividing D > 0. then (5.7) has at most two s~lut~ioris. Ljuriggren [133] proved t,hat when D = 7. (5.7) has no solutjioris when p is odd. More recently. J . H. E. Cohn [70] has shown that for 46 values of D 5 100, (5.7) has no solutions. H. Hasse [94] and F. Beukers [37] have written excellent surveys on (5.7). Beukers' thesis [37] and his two papers [38] and [39] contain substantial new results as well. In [38]. Beukers studied (5.7) for p = 2 and proved. among other things, a co~ljecture of Browkin and Schinzel on the number of solutions when D > 0. 111 his second payer [39], assuming that D is a negative int,eger and p is an odd prime not dividing D. he showed that there are a t most four solutions. hloreover. he gave a farnily of equations having exactly three solutions. 111 both papers. hypergeometric funhons play a central role. LI. H. Le has written several papers on (5.7). most of t,hern improving results of Beukers. To describe some of his results, assume in t,he sequel that D < 0, t,hat (D ,p ) = 1, p is prirne, and that N(D,p ) denotes the number of solutions of (5.7). In [124], Le considered the case when p = 2 and showed that, in various cases, N(D, 2) 5 2, 5 3. = 4. In [I231 and [125], Le considered the case of an odd prime p and showed that if max(D,p) is sufficiently large (which is made precise), then N ( D , p) 5 3. There are many further generali~at~ions of (5.7) studied by Le and others. but we will not discuss these here.

QI:ESTION 469 (JIMS 5, P. 159; 15, P. 97). The number 1 + n ! is a perfect squcre for the values 4,5 ,7 of n,. Find other values.

In volume 15. citing Dickson's History [73, p. 6811. RI. B. Rao wrote that the question was originally posed by H. Brocard [46] in 1876, and then again in 1885 [47]. (The problem was also nlentioned on t,he Norwegian radio program, Verdt aa vit,e (Worth knowing).) Furthermore. Dickson [73, p. 6821 reported t,hat A . G6rardin [83] remarked that, if further solutions of

exist. thcn r n has at least 20 digits. H. Gupta [88] found no solutions in the range 8 5 r~ 5 63 and thereby concluded that rrr has at least 45 digits. Recent calculations [35] have shown that there are no further solutions up to TL = 10".

In 1993. ill. Overholt [I521 proved that (5.8) has only finitely many solutions if the weak form of Szpiro's conjecture 1s true. but this remains unproved. To state the weak form of Szpiro's conlecturc. which is a special case of the ABC conjecture. first set

where p denotes a prime. Let a, b, and c denote positive integers, relatively prime in pairs and satisfying the equality a + b = c. Then the weak form of Szpiro's


conjecture asserts that there cxists a constant s slich that

(abc 5 N6 (abc).

For a furtlier discussion of Szpiro's conjecture. see a paper by S. Lang [119, pp. 44-45]. The more general qua t ion

was examined b j A. Dabrowski [72] in 1996. By a short. elementary argument. he proved that. if .4 is not a square, then there are only finitely many solutions of (5.9) in positive integers rn and n. He also showed that. if A is a square, then (5.9) has only finitely many solutions, provided that the weak form of Szpiro's conjecture is true.

The problem of finding solutions to (5.8) also appears in R. K. Guy's book [89, pp. 193-1941.

QL~ES~IION 723 (JILIS 7, P. 240; 10. PP. 357-358). If [L] denotes the greatest zntegrr In x . and n 7s any posztzve zntegrr, show that

(ii)

(iii) [A+ -1 = [J&q All three parts of Question 723 may be found on the first page of Rarnanujan's

third notebook 1171, vol. 2, p. 3611, and proofs can be found in Berndt's book [23, pp. 76-78]. A. A . Krislmaswami Aiyangar [2] later posed a problem giving analogues. one involving fourth roots and one involving fifth roots, of all three parts of Question 723. In a subsequent paper [3], he established theorems generalizing the results in his problem. a i d so further generalized Rarnanujan's Question 723. K . J . Cheii [60] has also established extensions of (ii) and (iii). Part (iii) appeared on thc M'illiam Lowd Putnam exam in 1948 [85].

QL ESTION 770 (.JIhIS 8. P. 120). If d(n) denotrs the r~urnher of dzvzsors of n (p.g., d(1) = 1, d(2) = 2 . 4 3 ) = 2, d(4) = 3 . . . . ) show that

is a convergent series; and that

(ii)

is n divergen,t series in th,e strict sense (2.e.. not oscillating )

There are two published solutions to Question 770. The first is by S. D. Chowla [63] and usc5 two thcorerns of E. Landau The second, by G. N. \Vatson [212], is cornpletelv different arid uses Dirichlct's well-known asymptotic formula

d(n) = .r. log .r + (2? - l).r + O(&) r r < r


(where 7 denotes Euler's constant), as 2 tends t o m, as well as similar formulas for the sum over odd n and for sums over n 5 1,3 (mod 4). due to T . Esterinam [76].

QUESTION 784 (JILIS 8. P. 159). If E(T) denotes fhe fmc t~ona l part of x (e.g. , F(T) = 0.14159.. . ), and zf N I S a posztme znteger. shoui fhat

J liin !VF(N&) = -; \-x 2 f i

(ii) lim log M ) ~ - J ' F ( N ~ ~ / " ) = 0, v - x

where n is any integer and p is any positive nmmber: show further that i n (ii) p cannot be zero.

Question 784 is concerned with the approximation of irrational nunlbers by rational riumbers. The proposed equalities should be compared with Hurwitz's theorem [150, p. 304, Thm. 6.111: Given any irrational number <. there are infinitely man; distinct rational numbers rrrln such that

The constant 1/& is best possible. To sec how Question 784 rclates to Hurwitz's theorem. we more closely exanline the first assertion in Question 784, which iinplies that there exists a sequence of positive integers {lV,:) tending to x such that

as NA --t cc. Setting A l k = [NA a], we may write the last formula in the equivalent form

which makes clear the relation t o Hurwitz's theorem. A partial solution was given by A. A. Krishnaswarny Aiyangar [I]. and a com-

plete solution was fourld by T. Vijayaraghavan and G. N. Watson [205].

QUESTION 427 (JIMS 3, P. 238; 10. P P . 320-321). Express

(A.r2 + Bzy + py2)(Ap2 + B p q + c ~ ' ) In the form Au' + B u u + C ~ Z , ~ : and hence show that. zf

then one set of the values of 11 and 1 1 zs

Question 427 appears in Rarnanujan's second notebook [171, p. 2661. In Berndt's book [23, pp. 9-10] it is shown that Question 427 is a consequence of


a more general lemma. The two published solutions in the Journal of the Indian Mathematical Society are more complicated.

In fact, Question 427 is a special case of Gauss' theory of composition of binary quadratic forms [69, p. 2121, [51, Chap. 71, which we very briefly describe. Suppose that Ql (x, y ) and Q2(x, g) are integral, positive definite quadratic forms of discriminant d. Then if Q:j is a form of discriminant d,

where, for cert,ain integral coefficients A,, B,, 1 < j 5 4,

Thus, Ramanujan considered a special case of Gauss' theory for Q3 = Q1 = Qn, and explicitly determined the coefficients A,, B,, 1 < j 5 4 for IL, 71 , or z:j, y3, in the notation above.

QUESTION 489 (JIMS 5, P. 200: 7, P. 104). Show that

As A. C. L. Wilkinson observed in volume 7, Question 489 gives the value of Weber's class invariant f(m) [219, p. 7231, which is also found (with different notation) in both Ramanujan's first and second notebooks [24, p. 1921. Recall [24, p. 1831 that, for each positive real number n, Ramanujan's class invariant G, is defined by

where q = exp(-.irfi). In the notation of Weber [219], G,, =: 2- ' 1 ' f ( f i ) . Weber [219] calculated a total of 105 class invariants, or the monic irreducible

polynomials satisfied by them, for the primary purpose of generating Hilbert class fields. Without knowledge of Weber's work, Ramanujan calculated a total of 116 class invariants, or the monic irreducible polynomials satisfied by thern. Not surprisingly, many of these had also been calculated by Weber. Using, most likely, methods considerably different from those of Weber in his calculations, Ramanujan was motivated by the connections of class invariant,^ with the explicit determina- tions of values of theta-functions and the Rogers-Ramanujan continued fraction. After arriving in England, he learned of Weber's work, and so when he wrote his famous paper on modular equations, class invariants, and approximat,ions to .ir [165]. 1172, pp. 23-39]. he gave a table of 46 new class invariants. Since Ramanujan did not supply any proofs in his paper [165], [172, pp. 23-39] or notebooks, Watson took up the task of calculating class invariants and wrote seven papers on calculating invariants. with three of them [214], [215], [216] specifically directed at verifying Ramanujan's class invariants. After Watson's work, a total of 18 of Ra- manujan's class invariants remained to be verified up to recent times. Using four distinct methods, Berndt, H. H. Chan, and L . 4 . Zhang complet,ed the verification of Ramanujan's class invariarits in two papers [28], [30]. This work can also be found in Chapter 34 of Berndt's book [24]. For expository, less technical accounts

THE PROBLEMS SUBMITTED BY RAMANUJAN 3 1

on Ramanujan's class invariants, their applications. and attempts to prove them. see two further papers by Berndt, Chan, and Zhang [31], [32]. Not all of Watson's verifications of Ramanujan's class invariants are rigorous. Zhang [222], [223] has given rigorous derivations of the invariants calculated by Watson by means of his "empirical method." while Chan [59] has taken Watson's empirical method, ern- ployed class field theory to put it on a firm foundation, and determined several new invariants as well.

QUESTION 699 (JIMS 7 . P. 160). Show that the roots of the equations

6) (ii)

can be expressed in terms of radicals.

First observe that

and that

Thus, Ramanujan's problem can be reduced to solving two quintic polynomials. It is doubtful that Ramanujan had actually solved these two quintic polynomials. It is unclear why Ramanujan introduced these two linear factors. In an unpublished lecture on solving quintic polynomials, Watson [218] remarked, "I do not know why Ramanujan inserted the factor x + 1; it may have been an attempt at frivolity, or it may have been a desire to propose an equation in which the coefficients were as small as possible, or it may have been a combination of the two."

Watson [211] observed that 2 p 1 / 4 ~ 4 7 , where G,, is defined by (5.10), is a solution of

a result which is also found in both Ramanujan's first and second notebooks [23, p. 1911. The class equation for G7y is not found in Weber's book [219]. However, on pages 263 and 300 in his second notebook [171], Ramanujan claimed that 21/4/G70 is a root of

This result can be deduced from equivalent results due to R. Russell [182] and later by Watson [217]. See also Berndt's book [24, pp. 193, 2751. Watson [211] furthermore pointed out that (5.13) was explicity solved by G. P. Young 12211 in a paper devoted to the general problem of explicitly finding solutions to solvable quintic polynomials and to working out many examples, the first of which is (5.13). Young remarks that (5.13) was "brought under the notice of the writer by a mathematical correspondent," whom Watson conjectured was A. G. Greenhill, who had also studied (5.13). A few years later, A. Cayley [58] considerably simplified Young's calculations. Since the solutions of (5.14) had not been given in the literature, Watson [211] explicitly determined them. The work of Ramanujan and Watson is summarized in a paper by S. Chowla [65].


D. Dummit [75] and V. M. Galkin and 0. R. Kozyrev [82] have also examined (5.13) and (5.14) and provided some insights into the Galois theory behind the two equations. For example, the Galois groups of the Hilbert class fields over Q are dihedral groups of order 10. Unaware of the work of Russell, Watson, and others, Galkin and Kozyrev rederived the class equation (5.14).

QUESTION 629 (JIMS 7, P. 40; 8, PP. 25-30). Prove that (5.15) l x - + x e-nn2x cos {nn2 JG?) = 2

n = l n=l

and deduce the following:

(ii)

In volume 8, three solutions are given. In the first, the solver erroneously

7L= 1 71= 1

Chowla [62] pointed out these mistakes and evaluated each series above in terms of gamma functions. The second and third solutions by N. Durai Rajan and M. Bhimasena Rao, respectively, are correct.

Equality (5.15) can be found in Section 23 of Chapter 18 in Ramanujan's second notebook [22, p. 2091, where it is a corollary of a more general transformation formula, Entry 23 [22, p. 2081. Part (ii) is given as Example (iv) in Section 7 of Chapter 17 in Ramanujan's second notebook [22, p. 1041. After Ramanujan, set

In that same Section 7, Ramanujan offers the values of p(eP"), p(epnJZ), and p(e-2") [22, pp. 103-1041, all of which are classical. In particular [22, p. 1031,

Observe that (i) can be written in the form

In view of (5.16), we see that (5.17) explicitly determines cp(e-"). Part (i) is found in the second notebook, where it is an example in the aforementioned Section 23 [22, p. 2101. In Berndt's book [22, p. 2101, one can find a short proof that uses the same generalized theta transformation formula that leads to a proof of (5.15).

Part (i)? in the form (5.17), is also recorded in Ramanujan's first notebook [171, p. 2851. The values p(e-:'"), p(e-7k), p(ep"), and p(e-4") are also recorded in


the first notebook and were first proved by Berndt and Chan [27]; see also Berndt's book [24, pp. 327-3281, where several "easier" values of p are also established 124, p. 325, Entry 11. Explicit values of cp yield at once explicit values for the hypergeometric function 2 ~ L ( $ , i: 1; k2) and for the complete elliptic integral of the first kind K(k) , for certain values of the modulus k [24, p. 323, eq. (0.4)].

We shorten Ramanujan's formulation of Question 584. in part, by using the notation

n-1

(5.18) (a; q)n := n(l - aqk), (a:q), = hrl(a;q),,. (ql < 1.

QUESTION 584 (JIMS 6' P. 199). Examine the correctness of the following results:

The identities (5.19) and (5.20) are called the Rogers-Ramanujan identities. and they have a long, interesting history, which we briefly relate here. As the word- ing of the problem intimates. Ramanujan had not proved these identities when he submitted them. Also. as their name suggests. they were, in fact. first discovered and proved by L. J. Rogers [180] in 1894. Ramanujan had evidently stated them in one of his initial letters to Hardy, for Hardy [172, p. 3141 later claimed that. "They were rediscovered nearly 20 years later by Mr Ramanujan, who communicated them to me in a letter from India in February 1913." (For a discussion of Hardy's assertion. see Berndt and Rankin's book [36, pp. 43-44].) Hardy in- formed several mathematicians about (5.19) and (5.20). but he obviously failed to consult with Rogers. The "unproved" identities became well-known, and P. A. MacMahon stated them without proof and devoted an entire chapter to them in volume 2 of his treatise Combznatory Analyszs [135]. In 1917. Ramanujan was perusing old volumes of the Proceedzngs of the London Mathematzcal Soczety and accidently came across Rogers' paper [180]. Shortly thereafter, Rainanujan [l7O], [172, pp. 214-2151 found his own proof. and Rogers [181] published a second proof. There now exist many proofs, which have been classified and discussed by G. E. Andrews in a very informative paper [6].

The Rogers-Ramanujan identities are recorded as Entries 38(i). (ii) in Chapter 16 in Ramanujan's second notebook [22, p. 161. It is ironic that they are. in fact. limiting cases of Entry 7 in the same chapter, as observed by R. A. Askey [22, pp. 77-78]. Entry 7 is a limiting case of ViTatson's transformation for 8P7, and, in view of the complexity of Entry 7. it is inconceivable that Ramanujan could have found it without having a proof of it.

The Rogers-Ramanujan identities have interesting combinatorial interpreta- tions. first observed by ICIacMahorl [135]. The first. (5.19), implies that: The number of partitions of a positive integer n into distinct parts. each two differing by at least 2. is equinumerous with the number of partitions of n into parts that are congruent to either 1 or 4 modulo 5. The second implies that: The number of partitions of a positive integer n into distinct parts. with each part at least 2 and


each two parts differing by at least 2. is equinumerous with the number of partitions of n into parts that are congruent to either 2 or 3 modulo 5.

For further history and information on the Rogers-Ramanujan identities, see Andrews' paper [6] and books [4, pp. 103-1051, [5] and Berndt's book [22, pp. 77-79].

6. Integrals

After R.amanujan (e.g., see [22, p. 36, Entry 22(ii)]), set

Using this notation and the notation (5.18). we abbreviate Ramanujan's Question 386.

QUESTION 386 (JIMS 4, P. 120; 7, PP. 143-144). Show that

The evaluation (6.1) can be deduced from the theorem

which is stated by Ramanujan in his paper [168, eq. ( l g ) ] , [172, p. 571. The deduction of (6.1) from (6.2) is given on the following page in that paper [168, eq. (24)], [172, p. 581. Ramanujan evidently never had a rigorous proof of (6.2). for he wrote [168], [172, p. 571 "My own proofs of the above results make use of a general formula, the truth of which depends on conditions which I have not yet investigated completely. A direct proof depending on Cauchy's theorem will be found in Mr Hardy's note which follows this paper." (That paper is [go]. [91, pp. 594-5971,) The special case a = 0 of (6.2), which, of course, contains (6.1), is stated by Ramanujan in his Quarterly Reports to the University of Madras, the focus of which is Ramanujan's "Master Theorem." To see how Ramanujan deduced the special case a = 0 from his "Master Theorem," see Hardy's book [92, p. 1941 or Berndt's book [20, p. 3021. The solution of Question 386 by N. Durai Rajan in volume 7 employs a partial fraction decomposition of the integrand.

R. A. Askey [9]. [lo], [ll], [12] has shown that (6.2) is a q-analogue of the beta integral. For references to extensions and further related work, see the aforementioned papers by Askey. A nice discussion of (6.2) may also be found in his book [7, Chap. 101 with Andrews and R. Roy. The evaluation (6.2) also appears as Entry 14 in Chapter 16 of Ramanujan's second notebook. See Berndt's book [22, p. 291, where several references for (6.2) and kindred integrals can also be found.

QUESTION 783 (JIMS 8 , P. 159; 10, PP. 397-399). If


show that

This is a very beautiful result which has been greatly generalized by Berndt and R. J . Evans [34] in the following theorem.

THEOREM 1. Let g be a strictly increasing, differentinble function on [0, cx;) with g(0) = 1 and g ( w ) = x. For n > 0 and t > 0, define

Suppose that

converges. Then

4 7 1 ) + 4 l l n ) = 2 4 1 ) .

To deduce (ii) of Question 783, let g(t) = l+ t . Observing that v ( t ) = t ( l+t) ' lpl , setting u = 1 + t , and using the value p(1) = 7r2/12, we find that Theorem 1 reduces to (ii). The proof by N. Durai Rajan and "Zero" in volume 10 is longer than the proof by Berndt and Evans of the more general result. Question 783 can be found on page 373 of Ramanujan's third notebook [22, pp. 326-329, Entry 411.

QUESTION 308 (JIMS 3, P. 168; 3, P. 248). Show that

7r3 T (i) 0 cot 0 log(sin 0)dO = - - - - log2 2,

48 4

Observe that

which is called Catalan's constant. The arithmetical nature of C is unknown, and this is a long outstanding, famous problem. It is ~onject~ured that C is transcendental. The function

was studied by Ramanujan in his paper [166], [172, pp. 40-431 and in his notebooks [20, pp. 265-2671, For related results in the notebooks. set [20, pp. 268-273, 285-2901 and [24, pp. 457-4611, The function


was also examined by Ramanujan in his notebooks [20, pp. 264-265, 268, 285- 2881. In fact, (ii) is a special case of Entry 16 of Chapter 9 in the second notebook, which we can write in the form

1 sin(2nkx) rl (sin(nz)) = nx log 12 sin(rrx) + I

k = l k 2

(see [20, p. 264, penultimate line of p. 2851). Setting x = 114 in (6.4) and using (6.3), we deduce (ii).

For (i), observe that an integration by parts gives

which can be found in the tables of A. P. Prudnikov, Yu. A. Brychkov. and 0. I. Marichev [157, p. 544, formula 71. The solution by K. J . Sanjana in volume 3 proceeds similarly. Part (i) is Example 2 in Section 13 of Chapter 10 in Ramanujan's second notebook. and the proof given in [21, pp. 31-32] is quite different from the aforementioned proof. For values of other integrals akin to that on the right side of (6.5), see the aforementioned tables [157, pp. 543-5441.

QUESTION 295 (JIMS 3, P. 128; 5, P. 65). If ap = n, show that

Ramanujan's solution involving double integrals and Fourier transforms in volume 5 is very short and clever. His idea is examined in greater generality in his paper [168, Sect. 41, [172, pp. 53-58], where the example above and further examples are given. The identity (6.6) was communicated by Ramanujan in his first letter to Hardy [172, p. 3501, 136, p. 271. It is also found in Chapter 13 of Ramanujan's second notebook [21, p. 2251. In both the letter and notebooks, (6.6) is given to again illustrate the same general idea used by Ramanujan in his paper [168].

QUESTION 353 (JIMS 4, P. 40; 8, PP. 106-110; 16, PP. 119-120). If n is any positive odd integer, show that

and hence prove that

L - - - ('Lk+l).rr (2k+l) .rr 8 (2k + 1) ( C O S ~ , + cos - ~ T L )

The proofs of (6.7) and (6.8) by A. C. L. Wilkinson in volume 8 employ contour integration, but his proof of (6.8) is very long. In volume 16, Chowla showed that a much shorter proof of (6.8) can be effected by using the Poisson summation formula for Fourier sine transforms.


Related results can be found in Section 22 of Chapter 14 in Ramanujan's second notebook [21, pp. 278-2801. In particular [21, p. 79, Corollary], if cu,P > 0 with cup = 7rV4, then

L , ,

i = i l ( 2 t + 1) (cosh + cos d m )

QUESTION 463 (JIMS 5, P. 120). If

cos (nx) im e 2 n A - 1 dx = d n ) ,

then

Find cp(n), and hence show that

A complete discussion of these results can be found in Ramanujan's paper [169], [172, pp. 59-67] and in Berndt's account of Ramanujan's notebooks 123, pp. 296-3031.


for all positive integral values of n.

Wilkinson's solution in volume 8 uses contour integration.

7. Series

QUESTION 260 (JIMS 3, P. 43; 3, PP. 86-87). Show, without using calculus, that

Question 260 is the first question submitted by Ramanujan to the Journal of the Indian Mathematical Society. In fact, the first two questions were communicated by P. V. Seshu Aiyar, Ramanujan's mathematics instructor at the Government College of Kumbakonam.

The equality (7.1) is a corollary of Entry 4 of Chapter 2 in Ramanujan's second notebook [20, pp. 28-29]. Ramanujan was not the first to pose (7.1) as a problem; Lionnet [I311 offered (7.1) as a problem in 1879. The problem can also be found in G. Chrystal's book [67, p. 3221.


QUESTION 327 (JIMS 3, P. 209). Show that Euler's constant, namely [the limit ofl

when n is infinite, is equal to

the first term i n the n t h group being

1% reformulate Question 327. Let Ak = i(3" I ) , k > 0. Then

where y denotes Euler's constant. Evidently, a solution was never published in the Journal of the I n d ~ a n Mathematzcal Soczety. However, the problem appears as Entry 16 in Chapter 8 in Ramanujan's second notebook, and a proof can be found in [20, p. 1961.

QUESTION 723 (J IMS 7, P. 240: 8, PP. 191-192: 16, P. 121). Show that

11-1 1

71- 1

= C tan-' 1

(ii) tall-' ~ = I I (2n + 2k + I )& k=o ((2k + l ) ~ 5 ) ~ '

Alehr Chand Suri showed in volume 16 that (i) and (ii) are the special cases r = 1 and n: = 1/& of the identity

which can be readily established by induction. The original formulation of (ii) is incorrect.

Ramanujan recorded further identities for tan-' sums in Chapter 2 of his second notebook [20, pp. 25-40].

QUESTION 768 (JIMS 8. P. 119; 8, P. 227). I f

T H E PROBLEMS SUBMITTED BY RAMANUJAN

show that

for all positive values of x; and that

(ii)

for all negative values of x.

The sign of the right side of (ii) is incorrect in the original formulation. Part (i) can be found in Ramanujan's third notebook [23, pp. 399-400, Entry 301. A companion result is given on the same page [23, p. 399, Entry 291.

QUESTION 769 (JIMS, 8, P. 120; 9, PP. 120-121). Show that

X 1

X 1

log C n log n log(2n) n =2 n =2

The original formulation contains an obvious misprint. In volume 9, K. B. Madhava, M. K. Kewalramani, N. Durairajan, and S. V. Venkatachala Aiyar offered the generalization

where

Question 769 coincides with Entry ll(iii) in Chapter 13 of Ramanujan's second notebook [21, p. 2171.

QUESTION 387 (JIMS 4. P. 120). Show that

Although no solutions were published, (7.2) has been rediscovered several times in the literature. Its home is in the theory of elliptic functions, as Ramanujan himself indicated when he proved (7.2) in his paper [165, p. 3611, 1172, p. 341. However, Ramanujan was not the first to establish (7.2). The first mathematician known to us to have proved (7.2) is 0. Schlomilch [187], [I881 in 1877. Although not explicitly stated, (7.2) was also established by A. Hurwitz [loll , [I021 in his thesis in 1881. Others who discovered (7.2) include C. Krishnamachari [112], S. L. Malurkar [137], M. B. Rao and M. V. Ayyar [177], H. F. Sandham [184], and C.-B. Ling [130]. The discovery of (7.2) in Ramanujan's paper [I651 or notebooks [I711 motivated the proofs by Watson [207], Grosswald [87], and Berndt [19]. More precisely, (7.2) is an example in Section 8 of Chapter 14 in Ramanujan's second notebook. The discussion in Berndt's book 121, p. 2561 contains many further references.


Schlijmilch and several others cited above, in fact, proved a more general formula than (7.2). Let a, P > 0 with aP = r2. Then

The special case a = P = -ir of (7.3) yields (7.2). Equality (7.3) is Corollary (i) in Section 8 of Chapter 14 in the second notebook [21, p. 2551. There is a further generalization. Let a , P > 0 with ap = n2, and let n be an integer exceeding 1. Then

where Bk , k 2 0, denotes the kth Bernoulli number. Observe, by (7.3), that (7.4) is not valid for n = 1. The first proof of (7.4) known to us is by Rao and Ayyar [177]. Ramanujan also discovered (7.4), and it can be found as Entry 13 in Chapter 14 in his second notebook [21, p. 2611. If we set a = /3 = -ir in (7.4), assume that n is odd, and replace n by 2n + 1, we find that, for each positive integer n,

which is also in Ramanujan's notebooks [21, p. 262, Cor. (iv)]. Apparently the first proof of (7.5) is by J . W. L. Glaisher [84] in 1889. Many proofs of (7.4) and (7.5) can be found in the literature, and readers should consult Berndt's book [21, pp. 261-2621 for many of these references.

Analogues of (7.3) and (7.4) wherein negative odd powers of k appear in the summands are also very farnous. Although we do not state the primary formula here, it is generally called "Ramanujan's formula for C(2n + I)." It is recorded by Ramanujan as Entry 21(i) in his second notebook [21, pp. 275-2761, and there exist many proofs of it. See Berndt's book 121, p. 2761 for almost two dozen references.

Most of the proofs of (7.2)-(7.5) do not involve elliptic functions or modular forms. However, Berndt [19] has shown that all of the identities discussed above, and others as well, can be derived from one general modular transformation formula for a large class of functions generalizing the logarithm of the Dedekind eta-function.

QUESTION 358 (JIMS 4, P. 78; 7, PP. 99-101). If n is a multiple of 4 , excluding 0, show that

Although not stated, n must be a positive integer. In fact, (7.6) is originally due to Cauchy [57, pp. 313, 3621. Other proofs of

(7.6) have been given by Rao and Ayyar [178], Chowla [64], Sandham [185], Riesel [179], Ling [130], and K. Narasimha Murthy Rao [149].

THE PROBLEMS SUBMITTED BY RAMANUJAN 4 1

Question 358 has a beautiful generalization. Let cu, ,i3 > 0 with a0 = 7r2, and let n be a positive integer. Then

If n is even and cu = /3 = .rr in (7.7), then (7.6) arises. Equality (7.7) is Entry 14 in Chapter 14 in Ramanujan's second notebook [21, p. 2621, and the corollary (7.6) is recorded immediately thereafter. The first proof of (7.7) appears to be by Malurkar [I371 in 1925. Proofs have also been given by Nanjundiah [147] and Berndt [19, p. 1771.

Analogues of (7.6) and (7.7) exist for negative powers of 2k + 1 in the summands. Such results were also found by Malurkar [137], Nanjundiah [147], and Berndt [19]. These formulas, (7.6), and (7.7) can be deduced from the same general transformation formula; see [19] for details.

QUESTION 546 (JIMS 6, P. 80; 7, PP. 107-109, 136-141). Show that

(ii)

The evaluations (i) and (ii) are Examples (v) and (vi), respectively, in Section 32 of Chapter 9 in Ramanujan's second notebook [20, p. 2891. These and several other evaluations of this sort were derived in [20, pp. 288-2901 from the following two related corollaries given in the same section. If 1x1 < 1, then

If 1x1 < 7r/4, then

In turn, these corollaries may be deduced from Whipple's quadratic transformation for a well poised generalized hypergeometric function 3F2.

QUESTION 606 (JIMS 6 , P. 239; 7, PP. 136-141, 192). Show that

Question 606 gives the value of X2(& - 2), where


and where Li, ( z ) , n > 2, denotes the polylogarithm

The value of X2(& - 2) is recorded in a slightly different form in Chapter 9 of Ramanujan's second notebook [20, p. 248, Ex. (vi)]. The evaluation (7.8) is also in Lewin's book [129, p. 19, eq. (1.70)], but it is originally due to Landen [118] in 1780.

QUESTION 642 (JIMS 7. P. 80; 7, P P . 232-233). Show that

(9

(ii)

The origina 11 formulation of (i) is incorrect, as pointed out by M. B. Rao in his solution. Ramanlian made the same mistake when he recorded it as Example (ii) in Section 8 of Chapter 9 in his second notebook. Part (ii) in Question 642 is given as Example (i) in that same section [20, p. 2501. For other examples of this sort, see Catalan's paper [56].

We have slightly reformulated the next question.

QUESTION 700 (JIMS 7 , P . 199; 8 , P. 152). Sum the series

where, for k > 1, ( ~ ) k = C ( C + 1 ) ( ~ + 2) . . . ( c + k - 1).

The sum of the series is

The two published proofs are elementary. The first, by K. R. Rama Aiyar, uses Euler's elementary identity

n x(l - ak:+1)a1a2.. . ak = 1 - a 1 a 2 . . . anti. k=O

8. Continued Fractions

QUESTION 352 (JIMS 4, P. 40). Show that


In both the original formulation and Ramanujan's Collected Papers [172, p. 325). (i) has an obvious misprint.

Question 352 gives the values of the Rogers-Ramanujan continued fraction

when q = e-'", -ep", respectively. As the name suggests, R(q) was first studied by L. .I. Rogers in 1894 [180]. In his first letter to Hardy, Ramanujan stated both (i) and (ii) [172, p. xxvii], [36, p. 291. and in his second letter, he gave the value of ~ ( e - ~ " / & ) [172, p. xxviii], [36, p. 571. In both letters, Ramanujan wrote that R(eP"fi) "can be exactly found if n be any positive rational quantity" [172, p. xxvii], [36, pp. 29, 571. (We have quoted from the first letter; the statement in the second letter is similar but is omitted from the excerpts of Ramanujan's letters in the Collected Papers.) In both his first notebook [171, p. 3111 and lost notebook [173, pp. 204, 2101, Ramanujan offered several further values for R(q). On page 210 of [173], Ramanujan planned to list fourteen values, but only three are actually given. Since the lost notebook was written in the last year of his life, his illness and subsequent death obviously prevented him from determining the values he intended to compute. In several papers [159]-[163], K. G. Ramanathan derived some of Ramanujan's values for R(q). All of the values in the first notebook were systematically computed by Berndt and Chan in [26], while all the values, including the eleven omitted values, were established by Berndt, Chan, and Zhang in [29]. Moreover, in the latter paper, the authors demonstrated for the first time the meaning and truth of Ramanujan's claim that ~ ( e - " A ) "can be exactly found" when n is a positive rational number. More precisely, they used modular equations to derive some general formulas for R(e-"A) in terms of class invariants. Thus, if the requisite class invariants are known, R(e-"fi) can be determined exactly. A brief expository account of this work can be found in 1311. S.-Y. Kang [106] has proved a formula in the lost notebook [I731 that likely was used by Ramanujan to compute values of ~ ( e - " 6 ) .

QUESTION 541 (JIMS 6, P. 79; 8, PP. 17-20 ). Prove that

K. B. Madhava offered an informative discussion of this problem in volume 8. His solution employs Prym's identity [158] for the incomplete gamma function

w h e r e ( a ) o = 1 a n d ( a ) , = a ( a + l ) ( a + 2 ) . . ~ ( a + n - 1 ) , n ~ 1 , w i t h a = x = 1 / 2 , and tjhe continued fraction of Legendre [127, p. 5091

1 a 1 a + 1 2 a t 2 3 a real, x > 0,

; c + i + x + 1 + x + x + x + . . . '

with z = a = 112.


Question 541 is a special case of the first part of Entry 43 in Chapter 12 in Ramanujan's second notebook [21, p. 1661,

where x is any complex number outside (-a, 01. In turn, Entry 43 is a corollary of Entry 42 [21, p. 1651, which gives a continued fraction of Legendre [126] for certain generalized hypergeometric functions 1 Fl (a; b; x). More precisely, if n is a nonnegative integer, and x 6 ( -OO,~] , then

Lastly, we remark that Question 541 is closely related to a result communicated by Ramanujan in his first letter to Hardy [172, p. xxvii], [36, p. 281, and given also as Corollary 1 of Entry 43 in Chapter 12 of the second notebook [21, p. 1661, namely,

which Watson [208] found in Laplace's treatise on celestial mechanics [121, pp. 253-2563. However, the first rigorous proof is due to Jacobi [103].

QUESTION 1049 (JIMS 11, P. 120). Show that

sin(nx) dx (i)

- Jtr - 1 2 3 >

n + - - - x + x + x + . . . n + n + n + . . .

sin(irnx)dx -

1 l2 22 32 (ii) - - - - -

n + n + n + n + . . x + - - -

x + x + x +. . . The formulas above are valid for n > 0. If we set

and

then (i) and (ii) can be respectively written in the forms

and


Thus f and g are self-reciprocal with respect to Fourier sine transforms. The first solution to Question 1049 was given by E. G. Phillips 11541. In his

proof of (i), Phillips used (8.2), and in his proof of (ii). he utilized the continued fraction

1 l2 22 32 d t = - - - - x > 0,

a special case of a continued fraction of T. J. Stieltjes [195]. L. J. Lange [I201 independently found a similar solution. (It is curious that immediately following Phillips' paper is the obituary of M. J. M. Hill, the first mathematician whom Ramanujan wrote from India [36, pp. 15-19]. Hill did not fully understand Ra- manujan's work, and there is no mention of Ramanujan in the obituary.)

W. N. Bailey [17] generalized (8.2) and thereby generalized (i), with, inter alia, the Bessel function J, (xt), - 1 < u < 312, appearing in the integrand. He gave a simpler proof of this result in a later paper [18].

9. Other Analysis


and prove from first principles that (b) = (a)

Question 261 was communicated to the Journal of the Indian Mathematical Society by P. V . Seshu Aiyar.

The more general product

was studied by Ramanujan in his paper [167], [172, pp. 50-521. In Chapter 2 of Ramanujan's second notebook, the product evaluations (a) and

(b) ran be found as Examples 3 and 4 in Section 11 [20, p. 411. In Chapter 13 of his second notebook [21, pp. 230-231, Entry 271, Ramanujan evaluated the product

where n is an even positive integer. Further product evaluations of the same kind are located on pages 279 and 287 in the second notebook [23, pp. 335-337, Entries 1-41,

QUESTION 571 (JIMS

show that (G)


This result was proved by Ramanujan in his paper [166], [1?2, pp. 40-431. An equivalent formulation can be seen on page 286 in his second notebook [24, p. 4611.

QUESTION 294 (JIMS 3, P. 128; 4, PP. 151-152). Show that [if x is apositive integer]

where 8 lies between $ and i. In volume 4, Ramanujan gave only a partial solution of this ultimately fa-

mous problem. A variant of Question 294 appears in Section 48 in Chapter 12 of Ramanujan's second notebook [21, p. 1811. The problem was completely solved independently by G. Szego [199], [ZOO, pp. 143-1521 in 1928 and by Watson [209] in 1929. In his first letter to Hardy [172, p. xxvi], [36, p. 271, Ramanujan claimed the stronger result

where k lies between & and A. Partial evidence for this claim arises from the asymptotic expansion [21, p. 1821

as x tends to m. In his paper [209], Watson remarked, "I shall also give reasons, which seem to me to be fairly convincing, for believing that k lies between & and - ." To the best of our knowledge, however, it appears that this claim has never been firmly established. However, K. P. Choi [61] has shown that, for all x 2 1, o < + + & .

Question 294 has the following connection with probability [139], [140]. Sup- pose that each of the n independent random variables X k , 1 < k < n, has a Poisson distribution with parameter 1. Then S,, := C;=I XI , has a Poisson distribution with parameter n. Thus,

Upon applying the central limit theorem, we find that

lim P(Sn < n) = i. r L - 3 s

For further connections to probability, see papers by K. 0. Bowman, L. R. Shenton, and G. Szekeres [43], D. F. Lawden [122], E. S. Key [lo?], and P. Flajolet, P. J . Grabner, P. Kirschenkofer, and H. Prodinger [81].

Question 294 is also related to the famous "birthday surprise problem;" see papers by M. L. Klamkin and D. J. Newman [108], and by M. Blaum, I. Eisenberger, G. Lorden, and R. J . McEliece [40].

E. T. Copson [?I] considered the analogous problem for e p x . For further ramifications of Question 294, including applications and analogues, see papers by 3. C. W. Marsaglia [138], J. D. Buckholtz [50], R. B. Paris [153], L. Carlitz [54], and K. Jogdeo and S. M. Samuels [104].


C. Y. Yilbrim 12201 has studied the zeros of the partlal sum in Question 294; a certain s l m involving these zeros arises in formulas for certain mean values of I<(; + zt) lL, where < denotes the Riernanri zcta-function.

Results related to Question 294 can be found in D. E. Knuth's book [log, pp. 112-1171, For further discussions of this problem see the commentary in Szego's Collected Papers [200, pp. 151-1521 and Berndt's book [21, pp. 181-1841.

QUESTION 738 (JIMS 8, P. 40). If

show that ~ ( x ) = 1 when x lzes be tw~en 0 and 1 ; and that *(x) # 1 when x > 1. Fznd the lzmzt of

{9(1 + c) - d l ) } 1 6

as t + 0 through posztive values

The first part of Question 738 can be found as a special case of a corollary in Section 13 of Chapter 3 in Rarnanujan's second not,ebook 120, p. 701. Rarnarlujari also communicated a version of this corollary in his Quarterly Reports to t,he Uni- versity of hladras [20, p. 306, eq. (1.14)]. In fact, (9.1) has a long history. It can be traced back t,o papers of J. I]. Lanibert [I151 in 1758. J . I,. Lagrange 11141 in 1770, and Euler [78]. [BO] in 1783. The most common proof utilizes the Lagrange inversion formula. Indeed, in P6lya and Szego's t,reatise [156, pp. 125, 135, 301, 316-3171, (9.1) is given as a problem to illustrate this formula. For many further references, see Berndt's book [20, pp. 72, 3071.

The first complete solution t,o Problem 738 was given by Szego [199], [200, pp. 143-1521. The limit queried by Ramanujan equals -2.

F. C. Auluck [13] and Auluck and Chowla [14] conjectured that ~ ( c ) is con- pletely rnonot,onic for x > 1, that is, (-l)"(k)(x) > 0, k = 0 , l . 2 . . . . . S. LI. Shah and U. C. Sharma [I911 established this inequality for k = 0.1,2,3,4. On the other hand, Shah [190] proved that xd(x) and e . r b ( x ) are not completely monot,onic. Finally. R.. P. Boas. Jr. [41] proved that $(.c) is not completely monot,onic on any interval ( c , m) .

QUESTION 526 (JIMS 6, P. 39). If n zs any posztzve quant7ty. show that

and find approximately th,e dzfference when n is great. H e m e show that

by approxzmately 1 0 ' I"

The editors of Ramanujan's Collected Papers slightly reworded the problem. Using Question 738, Szego [199], [200, pp. 143-1521 established the inequal-

ities posed by Ramanujan in Question 526. If we let A,, denote the difference of the right and left sides in (9.2). Szego claimed to have shown that

A,,,,),, < 1 0 ' " ' """' = 1.0451 x 10- ' ' O .

48 B R U C E C . B E R N D T , Y O U N - S E O C H O I , A N D SOON-YI K A N G

It was pointed out by S. S. hlacintyre [134] that Szego actually showed that

illacintyrc [134] improved Szego's result by proving that

correct to 5 significant figures. hlore generally. she proved that

QI-ESTION 7 4 0 (JIMS 8 , P. 40; 8 , P P . 220-221). If

where [XI denotes the greatest integer i n x . show that ~ ( x ) is a continmous function of z for all positiue values of .r, and oscdlates from +T to ,when s becomes infinite. Also diflerentmte ~ ( - r . ) .

where [ t] denotes the greatest integer in t , show th,at

Questions 730 and 753 are very closely related. with Stirling's formula playing a key role in the solutions.

QI~ESTION 7 5 4 (JIil lS 8 . P. 8 0 ; 12, P. 101: 13 , P. 1 5 1 ) . Show that

where E lies between and for all posltine ,ualwe.s of s.

K. B. ilZadhava's partial solution in voluirie 12 does not yield the bounds for E propostd b) Ranlanujail. In volume 13. E. H. Neville and C. Krishnainachary pointed out a couple of numerical errors in hladhava's solution, arid consequently hlacthava's hounds for E arc actually better than what hladllava originally claimed. h i t stdl not as sharp as those posed by Rainariujan. Neville arid Krishnamachary colicludc their remarks by writing. "hlr Rarnaiiujam's assertion is seen to be cred- ible. but more powerful means must be used if it is to be proved." The problem still appears to be open.

QUESTION 605 (JIILIS 6. P. 239; 7 , P P . 191-192). Show that. uihen x = m,

(.r+ a - b)!(X.r + 2b)!(9.r + a + b)!

(3.1- + n - r.)!(3.r + (1 - b + c)!(12.r + 3b)! = g.

This result is a straightforward application of Stirling':; forrriula.


On page 346 in his second notebook, Ramanujan stated a much more general result. Under certain prescribed conditions on m , n, &. B,. ak , and b,, 1 5 k 5 m , 1 1 3 1 n ,

Question 605 is easily seen to be a special case of (9.3), and is one of several examples offered by Rainanujan on page 346. For proofs of (9.3) and the aforementioned corollaries. see Berndt's book [23, pp. 340-3411.

10. Geometry

QUESTION 662 (JIMS 7 , PP. 119-120). Let AB be a diameter and B C he a chord of a circle ABC. Bisect the m,inor arc B C at M ; and draw a chord B N equal to half of the chord BC. .Join AAI. Describe two circles with A and B as centers and AAl and B N as mdi i , cutting each other at S and S', and cutting the given circle agazn at the points A1' and N' respectively. .Join A N and Biil intersecting at R, and also join AN' and BAT' intersecting at R'. Through B draw a tangent to the given circle, meetin,g AAl an,d AA1' produced at Q and Q' respectively. Produce AN and AllB to meet at P. and also produce AN' and AlB to meet at PI. Show that the eight points P , Q, R, S , S1, R', Q', PI are cyclic, and that the circle passing through these eight points is orthogonal to the given circle ABC.

This result appears in Entry 7(iv) of Chapter 19 in Ramanujan's second notebook, and a proof can be found in Berndt's book [22, pp. 244-2461, The problem was reproduced in Mathematics Today [174]. a journal for students of mathematics in Indian high schools and colleges. A total of 24 solutions were received.

QUESTION 755 (JIMS 8 , P. 80). Let p he the perimeter and e the eccentricity of a n ellipse whose center is C. and let CA and C B be a semi-major and a sema- rninor axis. From CA cut off CQ equal to CB, and also produce AC to P making C P equal to CB. From A draw AN perpendicular to CA (in the direction of CB). From Q draw QAl making uiith QA an angle equal to Q (which is to be determined) and meeting AN at A l . Join PA1 and draw PIV making with PA1 an angle equal lo half of the angle APM, and meeting AN at N . Wi th P as center and P A as radius descrihe a circle, cutting P N at K , an,d meetzng PB produced at L. Then, i f

a rcAL p - - - -

arc AK 4AiV' trace the changes In 4 when r varzes from 0 to 1. In partzcular, show that Q = 30" when e = 0; Q + 30" when e + 1 ; Q = 30" when e = 0.99948 nearly; Q assumes the mznzmum value of about 29"58f1 when e zs about 0.999886: and 6 assumes the

marzmum value of about 30O44:' when e as about 0.9589.

Question 755 appears as Corollary (ii) in Section 19 of Chapter 18 in Ramanu- jail's second notebook; a proof can be found in Berndt's book [22, p. 1901. LI. B. Villarino [206] has obtained irnproveinents for the approximatio~is given in the last part of the problem.

We arc grateful to Richard Askey, hlichael Bennett. David Bradlej-, Heng Huat Chan. David Dummit. Gergely Harms, Adolf Hildebrand, LIichael Hirschhorn.


Robert Larnphrrc. Jtrry Lange. Frank Olver, C. A. Rrddi. Bruce Reznick. Haakon IIkadcland. Keiineth S. Williams. arid the referee for helpful comments and refer- Cll('CS.

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I ~ E P A R T M E N T OF ~ I A ' I H E \ I A T I ( ' S . I .SI\ .EKSITY OF I L L I N O I S . 140!) VVEST ( ; I ~ E E N S T R E I ? ~ ' .

I -RHANA. T I . I . I N O I S 61x01. ITSA E-mazl address: [email protected] [email protected] [email protected]

Some Examples of Moment Preserving Approximation

B. Bojanov and A. Sri Ranga

Dedzcated to Professor L.J. Lnnge on his 70-th annzversary.

A s s l ' r r ~ c ~ . In this paper we give examples of rriornent preserving approximations produced by tlie generating functions and the kernel polynomials associated with orthogonal and similar polynomials.

1. Introduction.

Any squarr integrable function f ( x ) on a finite interval (a , h ) is completely drternliilecl by its moments

Therefore the moments are important characteristics of f . Then the problem of coiistructing an approximation of f of preassigned type that preserves as many monlerlts off as possible is of some interest. The general question of constructing an approximation from the linear space spanned by 72 given functions wo. w l , . . . , % I , , 1

that preserves twice as inany (that is, 2n) moments of a given function f with respect to another fixed system of functions p o , ~ ~ , . . . , Q , 1 , . . . was considered in

111. MTe give here some cxainples of moment preserving approximations spanned by

generating functions and kernel polynomials associated with the orthogonal polynomials and the orthogonal L- polynon~ials. The following proposition which is a specification of the rrlethod developed in [l] is the basis for the note.

THEOREM 1.1 . i l s surr~e that y 1 and y.~ are rectifiable curves ( i n particular, t in- ear segments) . A s s u m e th,at G(w, z ) i s a given integrable f imc t ion o n y, x 22. Let h(w) and g(z) be f i ~ n ~ t i o n s integrable respectively on yl and 72. A s s u m e also tha t the quadrature jorrnula

1991 Mnthematacs Subject Clnssificntaon. Prirnary 41A45; Secondary 41A55. This research was supportctl by the grants CNPq (300341/87-3) and FAPESP (96107748-3)

of Brasil.

58 B. BOJANOV AND A. SRI RANGA

is exact for the functions

preserves the moments p k ( f ) = Jy, f ( ~ ) $ ~ ( u ) ) h ( w ) d w , Ic function

, 171.

0 , 1 , . . . , m, o,f the

Proof. Assume that 0 5 Ic 5 m,. Using the assumptions of t,he theorem we get

= ll h ( w ) { o ( w ) j 4 * ( w ) d w = i i k ( O )

and the theorem is proved. As a particular case of this theorem we get the following.

COROLLARY 1.2. Let h ( w ) be a given zntegrable function on y. Assume that G ( w , t ) is any integrable function on y x [c, dl such that

is an algebrazc polynomial of degree < Ic i n t for each k = 0 , 1 , . . . , 2 n - 1 . Let g ( t ) be any non-negative weight function on [c, dl. Assume that {t ,); ' and {a , ) ; ' are respectively the nodes and the coeficients of the n-pomt Gaussian quadrature formula associated with the weight function g ( t ) on [c, dl. Then the approximation

preserue.s 2n m o ~ n e r ~ t s , p L ( f ) = j'? h ( w ) f ( w ) w h d w . k = 0 , 1 , . . . ,271. - 1. of the function

SOME EXAMPLES O F MOMENT PRESERVING APPROXIMATION 59

Proof. Assume t,hat 0 < k < 2 n - 1. Using the assumptions of the corollary we get

Since L, h(w)G(w, t )wkdw is a polynomial of degree < 2 n - 1 in t we can compute the integral with respect to t exactly by the Gaussian quadrature. Thus

and the corollary is proved. Remark: If G ( w , t ) . in addition to the requirements of the corollary. satisfies the

d condition G ( w , t )g(t)dt = 1. then it follows that o ( w ) preserves 2 n simple moments of the weight function h(w) . That is

In the following sections we give several examples of generat,ing functions and kernel functions that satisfy the requirements of the above corollary.

2. Generating functions.

Let Po(x ) , Pl (x), . . . be a system of orthogonal polynomials on [a, b ] , finite or infinite, with respect to a given weight function h ( x ) . Let

X

G ( s , t ) = c,, PI, (x) t l '

be the generating function of this system. For any polynomial Q ( x ) of degree < k we clearly havc

I1 since the Fourier coeficirnts b,, (Q) = h ( r ) Q ( x ) P,, ( x ) d s of Q with respect to the system {P,,) are equal to zero for n > I;. Thus G ( x , t ) satisfies the nlain conditions of corollary 1.2. As an immediate consequence the corollaries given below hold.


We have, see for examplr. [3],

where TI, ( x ) are the Tchebycheff polynomials of the first kind. Then

COROLLARY 2.1. Let F ( t ) be any non-negative function on [ - I , 11 which vanishes at most on a set of measure zero. Let { t , ) ; and {a,); ' be the nodes and coefJicients of the Gaussian quadrature rule associated with the weight function g ( t ) = ~ ( t ) / d m on [ - I , 11. Then the rational approximation

1 preserves 272 moments P L ( f) = L1 f ( x ) x L d x , k = 0 , 1 , . . . .2n - 1 , of the

functzon

For X > 0, let P;')(.c) be the Gegenbauer polynomials. It is known (see [3]) that they are orthogonal with respect to the weight function { 1 - x 2 ) ' p 1 / 2 in [-I. 11 and

Then corollary 1.2 implies

COROLLARY 2.2. Lrt { f , ) ; and { a , ) ; be the nodes and coeficzents of the Gauss- tan quadrature rule wzth resppct to the wezght g ( t ) = F ( t ) { l - tL)'- '/ ' on [ - I , 11. Then, for X > 0, the approxzmatzon

1 preserves 2 n moments p~ ( f ) = J", f ( x ) x L ( l - ~ ~ ) ~ " ' / ~ d r , k: = 0 , l . . . . ,272 - 1. of the functzon

Again, if X = 1, then a ( s ) is a rational approxirnatioii. Finally from e2'"' - - CT=C=o{H,, ( x ) / n ! ) t U . where H,, are the Hermite polyno-

mials. we obtain

COROLLARY 2.3. Let { t , ) ; and {a , ) ; be the nodes and coeficients of the Gauss- ian quadrature rule with respect to the weight F ( t ) e p " on ( - C G , ~ ) . Then the exponential polynomial

SOME EXAMPLES OF MOMENT PRESERVING APPROXIMATION 61

preserves 27-2, moments PI, ( f ) = J_Xlc f ( x ) x L e - " d x . k = 0.1, . . . ,2n - 1, of the functzon

3. Kernels from the Christoffel-Darboux sums.

\Ye now consider the kernel polynomials

where PI, are the orthonorrnal polynornials associated with the weight function h ( t ) in the interval [a, b].

T H E O K E ~ ~ 3.1. 6 % ~ . any n,on-negat,ive function F ( t ) the polynomial

where { t , ) ; and { a , ) ; are the nodes ond coeficzents of the Gausszan rule wzth respect to the wezght functzon g ( t ) = F ( t ) h ( t ) , preserves 2 n moments p k ( f ) =

J: h ( x ) f ( z ) r L d r . k = 0.1 , . . . ,271 - 1 . of the polynolnzal

where b J ( F ) = J(: h ( t ) F ( t ) P , ( t ) d t are the Fourier coefficients of F ( t ) .

Proof. It is seen that the function

L: h ( x ) K 2 ( x , t ) z L d x =

is a polynomial of degree < k . Then we can apply corollary 1.2 to conclude that a ( x ) preserves 271 moments of the function

which is a polynomial of degree N. This completes the proof.

Note that if N 5 272 - 1 t h m b,(F) = [ c : h ( t ) ~ ( t ) ~ j ( t ) d t = C: ' la ,PJ( t , ) , J = 0, . . . . X . Hcnce a ( x ) and f ( r ) are the same polynomial.

4. Generating functions of orthogonal L-polynomials.

Let y be a rectifiable curve symmetric about the origin. Let the function h(w). integrable on y. be such that h(-ul) = h (w) . If the sequence of rrionic polynomials {P,, (v,)} defined by


exists. then we may call it, in accordance with [3]. a sequence of orthogonal polynomials in relation to the function h ( w ) on y. These polynomials satisfy the recurrence relation

with Po(w) = 1, Pl (u:) = w and where a,,+, # 0 for n > 1. Assuming y n S = 0, where S = { w : %(w) = 0 and 15(u1)1 > m), it has

been shown recently in [2] that the monic polynomials B,, ( v ) . 71 > 0. given by

where w ( v ) = ( v - , 8 ) / ( 2 6 ) , a > 0 and p > 0, satisfy the properties

with B o ( v ) = 1, B l ( v ) = ZI - p. If v is a zero of B,, (v) then p2 / v is also a zero of

B7l ( v and

Here the curve F , which is the image of y by the mapping w + I ) , is such that for any v E l- also P"7: E I?. The function [ ( I ) ) = c ~ n s t . v - ' / ~ h ( w ( v ) ) , integrable in r, is such that

The sequence { v - ~ ( 1 z + 1 ) / 2 ~ B, ( v ) ) , which can be called a sequence of Laurent polynomials (or L-polynomials), have its elements orthogonal to each other in r with respect to the weight function [ ( v ) .

One interesting case for the curve y that we consider here is

y, = { W : R ( w ) = 0 and - < 5 ( w ) < m). The image of yl, under w ( v ) . is then the circle given by F 1 = { v : 7r = pelo, -71- < 0 < 7r).

Now let G ( w , t ) be a generating function for the sequence of monic orthogonal polynomials {P,, ( w ) ) such that

G ( w , t ) = c,, Pn ( w ) t n .

Then

is a generating function for B,, ( v ) and further

From (4.1) this last equation means that J, & ( u , t ) u"[(u)dv is a polyrlonlial of degree k in t . Now we can state the following theorem.


THEOREM 4.1. Let w ( v ) = ( v - 1 ) / ( 2 & ) . Then the real monzc polynomzals ( 1 B, ( u ) . n > 0 , where

~ ! , ' ) ( ~ ) = 2 ( 7 f i ) ~ ~ T ~ ( - i w ( 1 ~ ) ) , n > 1 ,

where C is the unit circle { u : u = e'" -K < 6' < K ) . The recurrence relation

( 1 1 wzth B,(,' ' ( v ) = 1 , B , ( u ) = ( v - 1 ) and B:' ' ( 2 1 ) = ( v - 1)'+2v, holds. Consequently

B $ ~ ' ) ( U ) = ll" + (-I)", n > 1.

Furthermore.

Proof. From the Tchebycheff polynomials TI, (ut). n 2 0, wp obtain that the real monic polynomials

P r L ( w ) = 2Pr'" ( z ) " ~ , , ( - i w ) , n > 1,

satisfy the orthogonality property

where yl = { w : % ( w ) = 0 and - 1 < S(w) < 1). From the recurrence relation and the generating function ~ ( " ) ( w , t ) = ( 1 - zu t ) / ( l - 2wt + t 2 ) for T, ( U I ) we obtain

X 2(2 - w t )

G(w, f ) = G(' I ( - iw, i t / 2 ) = = 2Po(w) -t C P,,(w)tn 4 - 4wt - t" rt=l

Hence the application of the transformation lc(z1) = ( 1 1 - /3) /(2&), with a = 0 =

1, gives the required results of the theorem. Now by considering the function G(' ) ( [ I , t / u ) we obtain from (4.2) and corollary

1.2 the following result.

COROLLARY 4.2. Let g ( t ) be m y n!on-negative function on the interval [c, dl. Let us also assume that 1 @ [c, dl . If {t,);' and {a , ) ; are the nodes and coeficients of the G a t m i a n quadrature rule associated with th,e weight function g(t) on [c, dl, then the rational approximation,

64 B. BOJANOV AND A. SRI XANGA

preserves 2n moments pr;( f ) = Jr. f ( v ) v " l / ( l + v ) ) d v , k = 0 , 1 , . . . , 2 n - 1, of the function

In the same way, now by considering the Gegenbauer polynomials piX). we obtain the following results.

THEOREM 4.3. Let w ( v ) = ( v - 1 ) / ( 2 & ) . Then for X > 0 the real monic polynomials B!Y") ( v ) , n > 0 , where

satisfy

where C is the unit circle { v : 2) = edB? -T < 0 < T ) . The recurrence relation

(G.X) with B,j"")(v) = 1 and B, ( v ) = ( v - I), holds. Furthermore,

Consequently

COROLLARY 4.4. Let g ( t ) be any non-negative function on the interval [c, dl. Let us also assume that 1 $ [c, dl. If { t , ) ; and { a , } ; are the nodes and coeficients of the Gaussian quadrature rule associated wzth the weight function g ( t ) on [c, dl, then the approximation

preserves 2n moments p k ( f ) = Jc, f ( v ) v k { ( v + 1 ) 2 X 1 / v ' ) d v , k = 0 , 1 , . . . ,2n - 1, of the function

5. Kernels from orthogonal L-polynomials.

We consider the rnonic polynomials B,, ( v ) . n 2 0. defined by

where ( ( 1 1 ) is e weight function such that the moments pn, = J, <(v)dv exist for

all rn - 0 , & I , f 2.. . . . It is known that when the hankel determinants H,(-") and


HA-"+') , n > 0 , associated with the moments are different from zero then the polynomials B,, ( v ) exist and satisfy

(5.2) Bn+l (v ) = ( v - Pn+l)Bn(v) - ~ n + ~ ~ B n - l ( v ) , 2 1,

with B o ( v ) = 1, B l ( v ) = v - PI, /31 = p ~ / p - 1 , and for n > 1

It is easily seen that po = p~ and p, = a,+lp,-l for n > 1. Polynomials such as B,(v) , when r C ( 0 , co), were first studied in [5] in the

context of the strong Stieltjes moment problem. Fa- more information about polynomials defined this way and the associated or+;~ogonal L-polynomials see for example [4], [6] and [7]. Note that the polynomials considered in the previous section are a particular case.

Now we consider the kernel polynomials

Using the recurrence relation (5.2) one can show that

(5.4) - K N ( v , t ) = B N ( u ) B N ( ~ ) + ( Y N + ~ P A T B N - I ( ~ ) B N - ~ ( ~ ) + Q N + ~ ~ N ~ ~ ~ N - z ( ~ , t ) ,

with K o ( v , t ) = 1 and ~1 ( v , t ) = B I ( v ) B l ( t ) + ~ 2 P 1 .

Proof. For k > N , the result is immediate from (5.1). For 0 5 k 5 N - 1, we obtain the result by proving the equivalent result

From (5.3) we have for any N 2 0 ,

For 1 < k < N - 1, from (5.1) and (5.4)

Now we are able to prove the result (5.5) by induction. We assume that F'LNP2)(t) = ( N - 2 ) pN-2tk for 1 5 k 5 N-3 . Since ~ i ~ - ' ) ( t ) = P N - ~ and H N P 2 ( t ) = p N - 2 B N - 2 ( t ) ,

( N - 2 ) the assumption leads also to F N _ , ( t ) = p N - z t N - 2 . With these results the relation

(5.6) immediatly gives F L N ) ( t ) = pNt"or 1 < k < N - 1. The initial conditions for the induct,ion are clearly seen. This completes the proof.

As consequences of this theorem we have the following results.


If t l , t 2 , .: . , t12 are distinct numbers then, for N > n and i = 1 , 2 , . . . , n . letting C k 1 clK.v(u . t , ) = 0 we obtain from

that K (u . t J ) . j = 1 ,2 , . . . , n are linearly independent. Here e, ( 1 , ) are the Lagrange polynomials associated with t l , t 2 , . . . , t,, . Also from the quadratic form

K\ ( t , , t , ) }" > 0. Hence, we obtain from corollary 1.2 1 ,=1

THEOREM 5 . 2 . Let the functzon F ( t ) be such that g ( t ) = t V $ ' ( t ) < ( t ) zs non-

negatrve zn [c. d] and the zntegral J~~ t v in lg( t ) f ( t )d t ezzsts for nl = 0,1, . . . . Let {t,);" and {a,);' be the nodes and coeficzents of the Gausszan rule wzth respect to the wezght g ( t ) . Then the polynomzal

preserves 2n moments, ph ( f ) = Jr 2 1 ' f ( P ) I ~ ~ < ( V ) ~ U . k = 0 , l . . . . .2n - 1, of the polynomzal

6. Rational moment preserving approximations.

As mentioned in [I], the problem of constructing a fimction from span {yo , y l ,

. . , ur1-1) that has given moments po. p1, . . . prL-l with respect to the system 0 0 , Q1. . . . ,4,, - I is trivially characterized by the determinant

here h ( x ) is a fixed weight function on [a , b]. Precisely, the linear system of equations

for k = 0 , 1 , . . . , rj - 1, in unknowns bo, b l , . . . , b,,-, has a solution if and only if D f 0. In other words, given the moments { p L ( f ) ) l ( of a certain function f with respect to po. 4,. . . . , @,, - 1, there exists a unique approximation of the form bo+o (x) + bl yl ( x ) + . . . + b,, _ $I,, - 1 ( x ) that preserves the monlents of f , provided D f 0. We shall consider in this section the particular case when { O o , @ I , . . . , 4,1- I }

is the algebraic system


and {vo, $,. . . . , yn-l } is the system of rational functions

where t l < t 2 < . . . < t,, are any fixed points which do not lie in [a, b]. We assume that these points are inside the interval [c, d l . where [a, b] n [c. d] = 0. Let us uote first the following sirnple fact.

L E M M A 6.1. For any weigh,t function h ( r ) o,n [a , b] we have

Proof. Assume the contrary Then tllcre exist numbers bo, b l , . . . , b , , 1 , at least one bJ distinct from zero, such that

This implies

And thus. there exists a non-zero rational function of the forrn P ( x ) / Q ( x ) , P E - ir , , -1, which is orthogonal to every polynomial of degree n - 1. Then P(rc)/Q(x) should have at least n sign changes in (a, b), which is clearly impossible. Thus Do # 0. The lemma is proved.

As an immediate consequence of this lemma and the remark at the beginning of this section, we deduce that for every function f and ever\ polynomial Q,, (I.) = ( x - ( x - t ) ( x - t,,) with zeros {t,}; lying outside [a. b] . there exists a rational function of the forrn ( .r) /Q, ,(x) with P,,-1 E xi,- 1 which has the same moments ,uo( f ) , pl ( f ) . . . . . p i l l ( f ) as f with respect to the algebraic systern 1, x , . . . , x l . Now c011sider the question: Does there exist a rational function Pn - ( x ) / Q T L ( x ) which preserves 2n moments of f ' If so. Pi, - 1 ( x ) is completely defined by Q, , ( z ) and consequently by the poles t l , t 2 . . . . . t,, . Indeed, P,,-1 ( x ) and Q,, ( x ) should satisfy the system

Since Q,, (x), sQ,, ( x ) , . . . , xri-I Q,, ( 2 ) can be written as linear corrlbinatio~ls of s'. J = 0 ,1 , . . . . 2n - 1, we get in particular that

h x s , r - , x h = 0, /c = 0 . 1 , . , n - 1.

which means that P,,_1 ( x ) is the polynomial of best L2-approximation of the function f ( s )Q, , ( x ) . Thus,


where {L,) is the system of orthonormal polynomials with respect to the weight h( t ) in [a, b] . Thus the computation of the moment preserving rational approximation Pn-1 (x) /Q,(x) to a given function f , if it exists, is reduced to an non-linear problem of determining the zeros t l , t a , . . . , t , of Q,(x). In view of theorem 1.1 such an approximation exists and it is completely determined if f can be presented in the form

with some integrable non-negative function g(t) on [c, dl. Indeed, in this case one can show as in the proof of lemma 6.1 that the functions

constitute a Tchebysheff system on [c, d] for each m. Then by theorem 1.1,

where {a,) are the coefficients and i t j ) are the nodes of the Gaussian quadrature formula for the system {Q,,)?-l on [c, d ] corresponding to the weight g(t) .

The question of convergence of P,- 1 ( z ) /Q , ( z ) to

as n tends to cc is of some interest. The estimation of the remainder n

can be simplified applying the following scheme. Given z @ [c, dl choose a function Q,(t) from span {%, . . . @ z n - l } such that

To determine Q, uniquely we can impose additional interpolation conditions on Q,,

say, of the form

at some points yl < . . . < yn from [c, dl. In particular one can choose y, = t,, j = 1,. . . , n, in which case Q, is the Hermite interpolant with double nodes at t l , . . . , t,. Having chosen @ we rewrite the remainder as

d d

9 ( f ) d t - ld Q,(t)g(t)dt + L @(t)g(t)dt = z - t


hIaking use of the fact that

(since the Gaussian quadrature formula for the Tchebycheff system Q o , . . . , @212- integrates exactly a) , and because interpolates 5 at t l . . . . . t,, . we get

Thus the problem of estimating the remainder is reduced to the estimation of L- approximation of the particular function A by interpolating functions from span

{@0, . . . 1 Q2ri-1).

Let us apply this scheme in the case [a. b] is replaced by a circle r of radius r > 1 and centered at z = 0. Choose [c, d] = [-I, I ] and h(x) = 1. Then we get

where

and p ( u ) is an algebraic polynomial. Clearly. by the Cauchy theorem, @ is a polynomial too. In fact,, in this case Q k ( t ) is simply equal to t" Now, by the Newton interpolation formula

where q ( t ) := ( t - y l ) . . . ( t - y,,) . Since

we get

- 1' g ( t ) Q r l ( t ) ~ ( t ) dt R ( f ; - Qr, ( z ) q ( z ) z - t

- - 1 '( ') - ' ( ' ) + Q, ( t ) --I 4 2 ) d t

z - t z - t

The integral of the first term in thc square brackets above is zero since ( q ( t ) - q ( z ) ) / ( z - t ) is a polynomial of degree n - 1 and thus the integral can be computed exactly by the Gaussian quadrature formula (which gives a value 0 since Q,, ( t ) vanishes at the nodes). Therefore the next elementary estimate

yields that / R( f ; z ) 1 tendh to zero as n tends to rx, for z far from [ - I , 11.

B. BOJANOV AND A. SRI RANGA

References

[I] B. Bojanov and L. Gori, Moment Preservzng Apprommatzons, Math. Balkanzca (New Series), to appear.

[2] C.F. Bracciali, J.hl.V. Capela and A. Sri Ranga, Symmetric orthogonal L-polynomials zn the complex plane, in Orthogonal Functions, Moment Theory and Continued Fractions: Theory and Applications, (W.B. Jones and A. Sri Ranga, eds.), Lecture Notes in Pure and Applied hlath. 199, Marcel-Dekker Inc., New York, 1998, 57-67.

[3] T.S. Chihara, An Introduction to Orthogonal Polynomzals, Mathematics and its Applications Series, New York: Gordon and Breach, 1978.

[4] W.B. Jones, 0. Njastati and W.J. Thron, Two-point Pad6 expansions for a famzly of analytzc functzons, J . Comput. Appl. hlath. 9 (1983), 105-123.

[5] Ur.B. Jones, W.J. Thron and H. Waadeland, A strong Stzeltjes moment problem, Trans. Arner. Math. Soc. 261 (1980), 503-528.

[6] A. Sri Ranga, Another quadrature rule of highest algebrazc degree of preczszon, Nurncrische Math. 68 (1994), 283-294.

[7] A. Sri Ranga, Symmetrzc orthogonal polynomials and the assoczated orthogonal L- polynomzals, Proc. Amer. Math. Soc. 123 (1995), 3135-3141.

DEPAK~XIENT OF ~IATHEMATICS, LNIVERSITY OF SOFIA, BLVD. J , u ~ ; s BOUCHER 5. 1126 SOFIA. BULGARIA

E-mazl address: borisQfmi .mi-sof ia.bg

DCCE. IBILCE. IJNI\.ERSIDADII ESTADUAL PAULISW (LTESP) . 15054-000 SXo Josf: DO

RIO PRETO. SP. BRAZIL E-mazl address: rangaQnimitz. dcce . ibilce . unesp. br

Contemporary Mathrrnatics Volume 236, 1'39'3

Relations between certain symmetric strong St ieltjes distributions

C. F. Bracciali

Dedzcated to L.J . Lange on the occaszon of hzs 70"' bzrthday

A B S T I ~ A ~ T . We consider strong Stieltjes distributions $ ( t ) that satisfy t" dq (p2 / t ) = - ( P 2 / t ) & d* ( t ) ,

for t t ( a , b ) C (0, m). 2w integer, 0 > 0, and the orthogonal Laurent polynomials related to ~ ( t ) . We study some relations that appear between these distributions when 2w is odd and 2w is cven. Further results concerning an example arc given.

1. Introduction

Connections between the theory of moments. continued fractions, Pad6 approxirnants and orthogonal polynomials are well known (see, e.g., Akhiezer [Ak] and Chihara [Ch]). Analogous relationships can be found in the study of two-point Pad6 approximants, continued fractions that correspond to two series. strong moment problems. and orthogonal Laurent (or L-)polynomials (see Cochran and Cooper [CC] , and Jones et al. [ JNT2, J T W ] ) . In this work we consider strong Stieltjes distributions, the related continued fractions and orthogonal L-polynomials.

Let 0 < /? < b 5 co, a = p L / b and $ ( t ) be a real, bounded and nondecreasing function on (a, b ) , with infinitely many points of increase in (a, b ) such that the moments

h

(1.1) p = ( ni = 0, *I, *2, ... ,

all exist. With these conditions $(t) is a strong Stieltjes distribution function on

( a , b) c (0, oc). We consider the sequences of mouic polynomials of degree n, B: ' ( z ) , n 2 1.

r = 0, *I, 5 2 , ... , defined by the conditions

1991 Mathematzcs Subject Classzficatzon. Primary 42C05, 33A65; Secondary 30R70 The author was supported by CNPq of Brazil Grant #200244/95-7.

@ 1999 Arnrrican Mathematical Society

71

72 C. F. BRACCIALI

The main properties of these polynomials can be found in a paper by Sri Ranga [ S r l ] . These polynomials are related to the orthogonal Laurent polynomials studied by Jones et al. [ J N T l , J N T 2 , J T W ] .

From the results given in [ S r l ] , it is easy to prove that the polynomials B:) (z ) . n > 0, r = 0, 1 1, k 2, ... , satisfy the three term recurrence relation

(.I ( r ) ( r ) with B:;')(z) = 1, B r r ) ( z ) = HI , > 0, a,+, > 0, n > 1. It is well known that the Stieltjes function given by

has the following asymptotic power series expansions

and

For any strong Stieltjes distribution, the Hankel determinants of order n, defined by HA") = 1, and

=

for n = 1 ,2 , ... and m = 0, +1, k 2 , ... , are positive. Hence there exist rational

P1 P2 Pn+r-1 fi+ll+_i+,+...+- Zn+r + lower other terms z z z J

B?) ( z ) - p P 1 - p P 2 z - . . . - p-n-rzn-T-l + higher other terms

when expanded accordingly. These rational functions form the two-point Pad6 table for the two series.

From the three term recurrence relation (1.3), we see that the polynomials B?' ( z ) , n > 0, r = 0, 1 1 , 1 2 , ..., are the denominators of the nth convergents of the M-fraction

The polynomials A ; ) ( Z ) are the numerators of the nth convergents of this continued fraction.

RELATIONS BETWEEN CERTAIN STRONG STIELTJES DISTRIBUTIONS 73

2. Comput ing t h e coefficients a$) a n d /3:)

We will display the coefficients of the three term recurrence relation (1.3), a?)

and p:), in the Q' - p table

The coefficients a:) and PC), n > 1, r = 0, f l , f 2 , ..., can be obtained from the moments p,, r = 0, f1, f 2, .. . using the quotient difference relations, for n 2 1,

with crj') = 0 and pir) = pr /p r l , for all values of r , (see McCabe [Mc]). ( - We now set Y~:) - pn + F'rom the equations (2.1), we see that

while

and

Substituting (2.3) in (2.2), we obtain

and

Then from (2.3),

These relations can also be derived from the algorithms given in a paper by Jones and Magnus [JM].

74 C. F. BRACCIALI

Hence, if the coefficients a:) and ~$1" are known for a fixed r then, from (2.1). we can calculate the coefficient ~ 1 ( ' + I ) as follows

p!r+l) = + & 1

We can calculate the coefficients a!,'") and p;;'"), for n > 2, using (2.4), then

( 7 )

(2.5) &+I' = ( T ) P n + %+I ( T j d:) + a:;i1

a n ( T ) + = P ,,) + a:? ' P,-, + CUP) ' The coefficients a:-'' and @c--ll', for n > 2, can be calculated similarly from

3. The Syw, P, b] distributions

We now consider the strong Stieltjes distributions, denoted by Syw, P, b] distributions, that satisfy the symmetric property

The moments for these distributions satisfy the equation

p7,, = p + i u " ~ - m - 2 w , m E Z.

These distributions were presented in the general case, S J [ w , P, b] distributions. in Bracciali, McCabe and Sri Ranga [BMS]. They have been studied by Sri Ranga et al. [S r l , SAM], for the cases when w = 112 and w = 0, where they are denoted by ScS(a, b) and ScS(a, b) respectively. They also were treated in an article by Common and McCabe [CM], for the particular case when w = 0 and = 1.

In [BMS] it has been shown that for an S J [ w , P, b] distribution, the related ( polynomials B," (z) , n > 0, r = 0, &1,52 , ... and the coefficients and ~ 2 ) .

n > 1, r = 0, 51. *2, .. ., satisfy some symmetric inversive properties, namely, for 2d odd and = $ - w,

and

for 1 = 0; *1,*2, ... . While for 2w even and j = -w

and

for 1 = 0> *l, f 2, ...


If d @ ( t ) is an S y w , 0, b] distribution, it is easy to show that, for m E 2,

(3 .6) tmd$( t ) is an ~ " w + m, P, b] distribution,

and

(3 .7) ( t + P)"%$(t) is an ~ " [ w + m / 2 , 0 , b] distribution

4. Some relations between the S ~ W , 0, b] distributions d$"( t ) and d y P ( t )

We now use the notation w" and dQO( t ) when 2w is odd and w' and d Q f ' ( t ) when 2w is even.

From the properties (3 .6) and (3 .7) we see that the following result holds.

THEOREM 4.1. Let d$" ( t ) be a n S3 [wO, P , b] distribution where 2w0 i s odd and d$"( t ) be a n S y w " , p, b] distribution where 2we i s even. They satisfy the equation

(4 .1) d $ " ' t ) = ( t + / ? ) d $ f ' ( t ) , t~ ( a , b ) ,

or the equation

(4.2) d q e ( t ) = t + U d @ ~ ( t ) , t t ( a , b ) ,

zf and only if

For the case 2w0 odd, then in (3 .2 ) , J = jO = - wO. While for 2w' even. in (3 .4 ) , j = j e = -w". Ram the above theorem wO = we + i, hence j" = j ' . We set J = j 0 = j e , to simplify the notation.

If we set

d+;(t) = tJ&,ho(t) and d$,"(t) = t Jd+" ( t ) ,

then. from the properties (3.6) and (3 .7) . we obtain

d$$(t) is an ~ " [ w " + j , P, b] = S J [ 1 / 2 , 0, b]

and

d$$ ( t ) is an S" [we + j , p, b] = s"[o, 0, b].

While from (4 .1) and (4.2), we see that

Since d$,"(t) is an S"1/2, /3, b] distribution and d$;(t) is an S 3 [ 0 , P, b] distribution, we may extend some results given in a paper by Sri Ranga and McCabe [SM] and in a paper by Sri Ranga [Sr2]. If we consider the distributions d+,"(t) and d~+$ ( t ) related by

d+y(t) = ( t + ( t ) ,

then the coefficients ,!$" and a${, n > 1. with respect to d$,"(t), and the coeff-

cients $,('I and a:$ n 2 1, with respect to d$$(t) are related as follows

where 1;:' = \ /76("/86(fl and 7 6 ( j ) = + ,(I) n + l , n 2 1.

76 C. F. BRACCIALI

They also are related by

,o(j)

n+l lp 1 , n > 1 and I!:) = 1. We need to calculate the coefficient where l k l l = - - 1P - 1

l p ) from @ J ) . On the other hand, if we consider the relation

t

then the coefficients pi( ' ) , u ~ y ! , ,B$) and a:?!, n > 1, are related by

,o(j) n+l lP where l g ) = -

( 3 + 1 , n > 1 and 1 f ) = 1.

ln-l + 1

5 . Example

We consider the S ~ W , p, b] distribution defined on ( a , b) ;

where 0 < /3 < b < CQ, a = P"b and 2w E 2.

THEOREM 5.1. For the S"w,@, b] distribution defined by (5.1), and for any

w > 112, some of the coeficients a t ) and p;), from the recurrence relation (1.3), are given by

= p, -

(5.2) a , f o r r > j , n > r + 2 w + 1,

( T I ,&) = P , an+, = a , f o r r 5 j , n > -r + 2,

where

and

1 - w , for 2w odd j = { w , for 2u, even.

It means that some of the coefficients a$) and p::) are constants and equal to a and p respectively, and they appear in a stair shape in the a - P table. It is easy to visualize this behaviour from some examples of the a - P table.


For example, for w = 112 and j = 0, some of the coefficients a:) and ,Or) related to the distribution

For w = 1 and j = -1, the distribution

yields the a! - table

I ( I That is

7 . ) d , ' )=P , a:, = a , f o r r > - 1 , n > r + 3 ,

P?) = P, a,+l = a, for r 5 -1, n -r + 2.

While for w = 3/2 and j = -1, the distribution

78 C. F. BRACCIALI

yields the a - P table

p ! ~ ) a ( r ) p r i p f ) Pri 3 4

( '1 ijjc) .. 2

That is

Similarly, for w = 2 and j = -2, the distribution

d $ ( t ) =

yields the a - f l table

That is

p p = 0, a;:) = Q, f o r r > - 2 , n > r + 5 , (1.1 P:;) = P , a,+, = a, for r 5 -2, n > -r + 2.

We now prove Theorem 5.1.

PROOF. For w = 112, in Sri Ranga [Srl] we find that

/3!,")=P, n > 1 , a f i = 2 a and 71 + 1 = a , n > 1 .

From the equations (2.5), we see that for r > 0 and n 2 r + 2 ,

@:) = /3 and a$) = a.


While from the equations (2.6) , we see that for r < 0 and n > -r + 2,

/3:) = o and = a .

Hence, the result (5.2) holds for i~ = 112. To show the result for w = 1 , we use the relation

d q O ( t ) = ( t + P)d$" ( t ) .

I We set

d$"( t ) = 1

d t , w" = 112 Jb-tJt-a and

Since we know that p;(C = p, n 2 1 , = 2 a , and a,,+, 40) = a , n 2 1 ,

then using (4.5) , we obtain

where 1 = d m . This last result can be found as an example in Sri Ranga and McCabe [SM].

We also know that for w = 1,

We then conclude that for w = 1 ,

p:,-l) = p p) r l + l - - a , n 2 3.

From the equations (2.5) , (2.6) and the above coefficients &') and a!,'), we can see that

PA' ) = P, a:,') = n , for r > -1, n > r + 3, ( r ) ( T ) = P , a,,+, = a , f o r r < -1, n > -r+2,

hence. the result (5.2) holds for w = 1. To show the result for w = 312, we use t,he relation

d$" ( t ) = - ( t + P ) d q ( f ) . t

80 C . F. BRACCIALI

We now set

and t

d$O(t)) = t+/3 d+'(t) =

Since we know the coefficients p:' obtain

pi-I' = p,

0'-1' = p Z , a:;:) = , n > 3,

where 1 = d m . Again using (2.5) and (2.6) we can see that

p r ) = p, a:) = a , f o r r > - 1 , n > r + 4 ,

~ 2 ) = P , crril = a, for r < -1, n > -r + 2,

and then conclude that the result (5.2) holds for w = 312. Similarly we prove the result for w = 2 , 5 / 2 , 3 , 7 / 2 , . .. .

From Theorem 5.1, we see that the M-fraction (1.4), for the distribution defined by (5.1) with w > 112, can be written as

The function

represents the limit of the periodic continued fraction f f z f f z f f z - - - z - p - z - p - z - p - . . .

for k = r + 2w when r > j and k = -r + 2 when r 5 j , with j = - w when 2w is odd and j = -w when 2w is even. From (5.3) we obtain

Then the continued fraction is equal to the rational function

A L ) ( z ) - g ( z ) A r l 1 ( z )

~ f ) ( z ) - g ( z ) B ~ J ( z ) ' for k = r + 2w when r > j and k = -r + 2 when r < j .

When we set w = 0 in the distribution (5 .1) , we obtain

(5.4) l + 6 l t dt .

d$ ( t ) = d m d z


THEOREM 5.2. For the S 3 [ w , P , b] distribution defined by (5.4), some of the coeficients a t ) and P?), from the recurrence relation (1.3), are related by

for r even, and

for r odd.

The a - P table for the distribution (5.4) is given by

In Sri Ranga and Bracciali [SB] it was proved that

where

(5.8) lo = 1, l l = - 2a1' + 1 , and 1,=1+ Q I P , n 2 2 .

lo + 1 1 + ln-1

In this special case

From the relations (3.5) with j = 0 and 1 = 0, we obtain

(5.10) 0:) = PL ,$) n+l = + ( 1 ) n L 1 . 1,- 1

PROPERTY 5.1. The coeficients l,, n 0, defined by (5.8) or (5.9), satisfy

where 1 = Jl+(ylP. W e define 1:+, + 1n+21n (1:+, - 1 ) = 1, for n < 0.

82 C. F. BRACCIALI

PROOF. For n = 0, the result follows directly from (5.9). For n > 0, from t,he relation (5.8), we have

After some simple manipulations with lz+l + ln+21n (L;+, - l ) , we obtain the result. 0

We now can prove Theorem 5.2.

PROOF. From the relations (2.5), we construct the row with superscript (2) in the cu - p table using the row with superscript (1) given by (5.10), we then obtain

(2) - (1) (1) P1 - P1 + 0 2 ,

Hence, using Property 5.1, we obtain

and

0;;) = p p , = Qp, n > 3.

Similarly, using the relations (2.5) we obtain the result for the rows with superscript

(3),(4),(5),... . From the relations (2.61, we construct the row with superscript (-1) in the

cu - table using the row with superscript (0), given by (5.7), we then obtain

Hence, using Property 5.1. we obtain

and (-1) - (1) P ~ = P : ) , a,+, -.,,+,, 7223 .

Similarly. using the relations (2.6) we obtain the result for the rows with superscript

(-21, (-3), (-41, ... . 0


In addition, for the distribution defined by (5.4): the coefficients a f i l and at:i, satisfy the property

We can prove this result by substituting (5.7) and (5.10) in (5.12), that is, for TL > 1)

Now using the relations (5.8), we obtain (5.12).

Acknowledgement. The author is grateful to the referee for the helpful suggestions, resulting in improvements to t,he initial version of this paper.

References

[Ak] N.I. Akhiezer, The classzcal moment problem and some related questzons i n analyszs, (translated by N. Kemmer), Oliver k Boyd, Edinburgh/London, 1965.

[BMS] C.F. Bracciali, J.H. McCabe, A. Sri Ranga, O n a symmetry zn strong dzstrzbutzons, in Con- tinued Fractions and Geometric Function Theory, (L. Lorentzen. 0. Njastad, and F . Ronning, eds.), J . Comput. Appl. Math., t o appear.

[Ch] T.S. Chihara, An zntroductzon to orthogonal polynomials, Gordon & Breach, New York, 1978.

[CC] L. Cochran, S. Clement Cooper, Orthogonal Laurent polynomials on the real line, in Conti- nued Fractions and Orthogonal Functions, Theory and Applications, (S. Clement Cooper and W.J. Thron, eds.), Lecture Notes in Pure and Appl. Math., 154, Marcel-Dekker, New York, 1994, pp. 47-100.

[CM] A.K. Common, J.H. McCabe, The symmetric strong moment problem, 3. Comput. Appl. Math 67 (1996), 327-341.

[JM] W.B. Jones, A. Magnus, Computatzon of poles of two-pant Pad6 approximants and thew lzmits, J . Comput. Appl. Math. 6 (1980), 105-119.

[JNTl] W.B. Jones, 0. Nj&tad, W.J . Thron, Two-point Pad6 expanszons for a famzly of analytzc functzons, J . Comput. Appl. Math. 9 (1983), 105-123.

[JNTZ] W.B. Jones, 0. NjLstad, W.J . Thron, Orthogonal Laurent polynomials and the strong Hamburger moment problem, J . Math. Anal. Appl. 98 (1984), 528-554.

[JTW] W.B. Jones, W.J. Thron, H. Waadeland, A strong Stzeltjes moment problem, Trans. Amer. Math. Soc. 261, (1980), 503 528.

[Mc] J.H. McCabe, A formal extension of the Pad6 table to include two-poznt Pad6 quotzents, J . Inst. Math. Appl. 15 (1975), 363-372.

[Srl] A. Sri Ranga, Another quadrature rule of hzghest algebrazc degree of preczsion, Numer. Math. 68 (1994), 283-294.

[Sr2] A. Sri Rangs, Companzon orthogonal polynomzals, J . Comput. Appl. Math. 75 (1996), 23 33.

[SAM] A. Sri Ranga, E.X.L. de Andrade, J.H. McCabe, Some consequences of a symmetry i n strong dzstrzbutions: J . Math. Anal. Appl. 193 (1995), 158-168.

[SB] A. Sri Ranga, C.F. Bracciali, A contznued fractzon assoczated with a special Stieltjes functzon: Comm. Anal. Theory of Contin. Fractions 3 (1994), 60-64.

[SM] A. Sri Ranga, J .H. McCabe, On pairwise related strong Stieltjes dzstrzbutions, Det Kongelige Norske Videnskaders Selskab 3 (1996), 3 12.

DEPARTAMENTO DE C I ~ N C I A S DE COMPUTA~AO E ESTAT~TICA. IBILCE-CNESP, 1505,1-000, SXo Jos i DO RIO PRETO, S P , BRASIL.

Current address: School of Mathematical and Computational Sciences, University of St Andrews, North Haugh, St Andrews, KY16 9SS, Scotland, UK.

E-mazl address: cleonicet3dcce. ibilce . unesp. br

Contemporary Mathematics Volume 236, 1999

Estimates of the rate of convergence for certain quadrature formulas on the half-line

Adhemar Bultheel, Carlos Diaz-Mendoza, Pablo Gonzhlez-Vera, and Ramon Orive

! Dedzcated to Jerry Lange on the occaszon of hzs 70th birthday

ABSTRACT. We investigate the rate of convergence of so-called n-point Gauss type quadrature formulas to integrals of the form f (x)da(x) where cu is a general distribution function on [0, m) and where f is analytic and admits a Laurent expansion in @ \ (0). The general results are then applied in the special case where dcu(x) = xu exp{- (xY + xpY))dx, a E E% and y E (1/2, m).

I 1. Introduction

Let a be a general distribution function on [0, co). We consider quadrature formulas of the form Ckl A,, f (x,,) with positive weights A,, > O and with real knots x,, > 0. If these quadrature formulas are exact for all integrals I , ( f ) =

h,"" f (x)da(x) where f is an arbitrary Laurent polynomial of the form Cy=Pp a,xJ where p = p(n) and q = q(n) are nonnegative integers such that p+q = 2n- 1, then these are called n-point Gauss type quadrature formulas. If we want to investigate the behavior of these quadrature formulas as n + m with both p ( n ) and q(n) tending to co, then it is natural to consider integrals ST f (x)da(x) where f (z) =

k fkz%uch that xr="=, fkz and Cp=l fPkzk represent entire functions. This class generalizes in a natural way the Laurent polynomials.

In this paper we shall give rates of convergence for these n-point Gauss-type quadrature formulas. In Section 2 we introduce the necessary notation and definitions. The main results are introduced in Section 3 and in Section 4 we apply these general results to the special case when da(x) = xa exp{-(x? + x-Y))dx, a E IR and y E (112, co).

1991 Mathematics Subject Classijicatzon. Primary 65D30; Secondary 41A21. The work of the first author was performed as part of the project "Orthogonal systems and

their applications" of the FWO under grant #G0278.97. The work of the other three authors was supported by the scientific research project of the

Spanish D.G.I.C.Y.T. under contract PB96-1029.

@ 1999 American Mathematical Society

85

86 A. BULTHEEL, C. D~AZ-MENDOZA, P. GONZALEZ-VERA, AND R. ORIVE

2. Nota t ion a n d definitions

Let a ( x ) be a distribution function on [0, m), i.e. a real valued, nondecreasing function with infinitely many points of increase such that the moment function

exists and is finite in the Lebesgue-Stieltjes sense. Define the moments

For simplicity, we assume that co = 1. Denote

whenever the Lebesgue-Stieltjes integral is defined and finite. Let { ~ ( n ) ) , , , ~ be a sequence of nonnegative integers such that 0 < p(n) < 2n - 1 for any n. Now, because of (2.1), all the moments ck, k E Z are finite and a polynomial Q,, of exact degree n (unique up to a multiplicative factor) can be constructed satisfying

Let {x,,);=, be the zeros of Q,, , then there exist positive weights {A,l,)~=:L=l such that

where IIk ( k > 0) denotes the space of polynomials of degree k at most. For p and q nonnegative integers, we write for subspaces of Laurent polynomials

where A is the space of all Laurent polynomials. Take L E A-,(,,),,(,; ( ~ ( n ) + ~ ( n ) =

2n - I), then, since L(z) = X-P(")P(X) with P E II2,,-1, one has

where A,,, = zyp)~,,, > 0, j = 1, . . . , n. The expression

will be called the n-point Gauss-type quadrature formula for da(x) in A_,,(,),,(,). Observe that I,,(L) = I,,(L) for any L E A_,(,) ,(,,). For further details about these quadrature formulas see [BDMGV097a] , [BDMGV097b] , JBDMGV0981.

In this paper we shall be concerned with the convergence of these quadrature formulas for functions admitting a Laurent expansion valid in C \ (0). That is we

RATEOFCONVERGENCEFORQUADRATURE 87

study the convergence of I n ( f ) when f is a Lebesgue-Stieltjes integrable function with respect to da(x) on [0, m ) which is of the form

f + (2) and f -(1/z) being entire functions. This class of functions f is clearly the most natural generalization of the Laurent polynomials.

We now introduce two functions R, and r, which we shall use to control the size of respectively the largest and smallest node x,, as n + m. These R, and r, are two arbitrary functions defined on [I, m ) which satisfy the following conditions: R, > 0 and r, > 0, R, is monotonically increasing and r, is monotonically decreasing, and

(2.5) lirn R, = cx and max{x,,, : 1 2 j < n) 5 R,, X'X

(2.6) lim r, = 0 and min{x,, : 1 < j < 7%) > rTL. x-X

Thus xjn E [rn,Rn], j = 1 , . . . ,n,. n E N. In the rest of this paper we also assume that the sequences ~ ( n ) and q(n) are

both nlonotonically increasing and that they satisfy

lim p(n) = lim q(n) = m. TZ"X n-x

Next, we introduce two monotonically increasing functions which will measure the degree of exactness of the formulas I,( f ). Set

(2.7) v(x) = min{n : q(n) 2 x), x > q(1)

and

(2.8) V* (x) = min{n : p(n) > x), x 2 p(1).

This gives the degree of exactness since

I,(x3) = Iu(J)(x3) and I,(xp3) = 1,*(3)(x-3), V j E N.

Assume that

lim p(n) -- = 8 E (0, I ) . n i x 2n

By (2.7) it immediately follows that

Hence

Since limn,, ~ ( n ) = m (recall that lim,,, q(n) = m), we get

Similarly, by (2.8) and (2.9), one obtains n

lirn --- = 6 . 2u*(n)


The functions which we introduce next, will enable us to formulate convergence results to be given in Section 3 in a more elegant form. These functions are

(2.12) - %n) $(8) = lim - n-co R n

2(1-8)

and

Furthermore we introduce

where

l l d n ) - in p(0) = lim -

n+m Ra 2(1-8)

and Q, the nth monic orthogonal polynomial for x-p(")da(x). We also need the so called Christoffel function of degree m for the distribution da,(x) = x-(p(")+')da(x) a s given by

IP(z) 1 2 K m ( z , a n ) = sup --- IIPII~ '

where IIPII: = I P ( x ) ~ ~ ~ ~ , ( x ) . This allows us to define

For any entire function f ( z ) = CEO f jz j , we define its order by

- logn-" p(f) = lim - "'" 1% Ifnl

and if 0 < p(f) < oo, its type is

- n J f n ( ~ ( f ) l " ~ ( f ) = lim , e = exp(1).

ep(f)

Finally, for s > 0, the indices CR and or are defined by

For more information and properties on several of the notions that were introduced, we refer to [Mha96]. There one can also find in greater detail several (polynomial) results that are related to some of the problems we discuss in this paper for the case of Laurent polynomials.


3. Rates of convergence

In this section we give the main results about the convergence o f the quadrature formulas I , ( f ) , n = 1,2 , . . . as defined in Section 2. To understand better the technical aspects, we start with two lemmas which give the relation between the parameters p, T and a . They are basically contained in [Lub83].

LEMMA 3.1. Let f + (a ) = x,& f;zj be an entire function.

a ) Suppose that for some q f > 0 and c+ > 0, R , < (c+x) 'J f for suficiently large x . Then

i ) p ( f + ) < l / q+ =+ a R ( f + ; s ) = 0 , V s > 0 4 +

ii) p ( f +) = l /q+ + a R ( f t ; s ) < (c+s e r ( f i ) / q + ) , V s > 0

b ) Suppose that for some q: > 0 and cf > 0, R , > (c;x)qf for suficiently large x . Then

i ) p ( f+ ) > l / q ; + a ~ ( f + ; s ) = cm, V s > 0 4:

ii) p ( f + ) = l /q: + o R ( f i ; s ) > ( c f s e r ( f t ) / q : ) , V s > O

REMARK 3.1. I f R , = ( K X ) ~ ' , then equality holds in a)ii) and in b) i i ) , so that aR( f + ; S ) can be estimated exactly in terms o f T ( f +).

The following lemma gives the analog for entire functions in l / z .

LEMMA 3.2. Let f - ( z ) = C,"=l f ~ z - j be an entire function.

a ) Suppose that for some q- > 0 and c- > 0 , l l r , 5 (c-x)q- for suficiently large x . Then

i ) p ( f - ) < l / q + a , ( f ; s ) = O , V s > O

b) Suppose that for some q, > 0 and c l > 0 , l l r , > (c,x)q; for suficiently large x . Then

i ) p ( f - ) > l / q , + a , ( f P ; s ) = m, V s > 0 -

ii) p ( f -) = l / q ; + a,( f - ; s ) > (c;s e r ( f - ) / q ; ) " , V s > 0

Now, taking into account that q ( n ) > 0, we have C,"=, X j , = co = 1. Thus, from the definition o f R, and r,, the following lemma can be deduced.

LEMMA 3.3. Let I , ( f ) = CGl X j n f (x,,) denote the n-point Gauss-type quadrature formula for I,( f ) in A-p(n).q(,), (p (n) + q (n ) = 2n - 1). Then

W e prove one more lemma.


LEMMA 3.4. Assume that f = f + f - is given as in (2.4). Furthermore suppose that Q ( 0 ) and I/$* ( 0 ) as given by (2.12) and (2.13) are both finite. Assume also that a R ( f T ; &)Q(O) < 1 and a , ( f - ; $ ) / + * ( 0 ) < 1. Then

PROOF. We have to prove that

Cx

and Ia [= fy z -3 ) =x f -~a(~-~) Cx 3

3=1

We shall only prove the second equality since the first one is proved in a completely similar way. By the dominated convergence theorem [Rud74, Theorem 1.341, it is sufficient to prove that CEl 1 f J - 5 - 3 1 is a Lebesgue-Stieltjes integrable function

with respect to d a ( x ) on [O, m). Since f - ( z ) = C,:, fJ-zPJ is holomorphic in

t? \ (0) where 6 = C U {m} is the extended complex plane, we can write

Because JbX x P J d a ( x ) 5 x - J d a ( x ) , b > 0, it results that

Thus we have to check that the series C,:, f J - I O ( x p J ) is absolutely convergent. Now by c) in Lemma 3.3, one has

l f ; l I I a ( ~ - ~ ) l = I f ; l l I u * ( J j ( x - 9 5 lf;lr;2(J)

To simplify the notation we set s = s ( 0 ) = a , ( f P ; $ ) / $ * ( O ) . (Recall that by assumption s < 1.) Thus, there exists some no such that for all n > no

Choose p > 0 such that T / = ~ ( 0 ) = ps = s + < 1. Then

Thus the series C3E1 fJPIa ( T - 3 ) converges absolutely and the lemma is proved.

RATEOFCONVERGENCEFORQUADRATURE 9 1

We are now in a position to state the first result concerning the estimate of the rate of convergence of the sequence I , ( f ) , n = 1 , 2 , . . . of Gauss-type quadrature formulas.

THEOREM 3.5. Let f ( 2 ) be a function satisfying (2.4) and suppose that ~ ( 8 ) and I /$*($) as given by (2.12) and (2.13) are both finite. Assume also that a r i ( f T ; &)+(O) < 1 and a , ( f p : $ ) / $ * ( H ) < 1. Then

where

PROOF. By (2.4), set f = f + + f - with f+(z ) = CJ>O=, fTz3 and f p ( z ) =

C,:, f;zp3. Thus, with E, = I, - I,,

E n ( f ) = E n ( f + ) - E n , ( f - )

By the previous lemma, we know that

Thus, since I n ( f ) = I , ( f ) , b'f E we can write IX: X

E n ( f ) = C f:En(x3) + C f [ E n ( ~ - ~ ) . 3=dn)+l 3=p(7~)+1

We study the convergence of both series in the right-hand side separately. First we consider the series x3Eq(n)+l f +E,(xj). From the definition of u ( x ) in (2.7), Lemma 3.3, and the monotonicity of R,, it follows that

sup 1 f j t l l " l I a ( s ~ ) - I T L ( d ) I 1 / ~ = sup 1 ~ ~ ~ I I ~ ( ~ ) ( Z J ) - I1 , (xJ) l l '~ 3>dn) 3>dn)

SUP I ~ ; ~ / ~ ( R : ( ~ ) + R ; ) ~ / J 3>4(71)

< 21/q(7L) SUP 1 -

3 > ~ ( 7 ~ )

Then

liln sup ~ ; I ~ / J I E ~ ( Z J ) ~ " - I < lim sup 1 f ; i 1 ~ ' ~ ~ ~ ! ~ ~ n -x

J > ~ ( T L ) 1 c - x

3 > q ( 7 ~ )

( 3 1 = lim sup ( f,+ll/%- - 11-X 3>dn) Z ( l - 0 ) R- L ( 1 - 8 )

Observe that since limn,, q(n) = cc and S U ~ ~ > ~ ( ~ , ) 1 fT ~ ' / J R ~ is monotonically 2 ( 1 - R )

decreasing. it holds that


We now proceed as in the proof of the previous lemma. By hypothesis, we know that a R ( f + ; &)+(0) < 1. Take p > 1 such that 77 = g(0) := p o ~ ( f + ; &)$(0) < 1, then there exists an no = n"(p) E N SO that for any n > no by (3.1)

and consequently

Thus, one sees that the series CZq(,)+, If; 1 1 I, (xj) - In (xj) 1 is convergent and it follows that

I, ( 2 n - x

5 7 7 l - O

j=q(n)+l

As the left-hand side in the last inequality does not depend on the parameter p, we can replace p by 1 so that

On the other hand, by assuming that a,( f -; $)/+*(8) < 1, and proceeding as above, it can be shown that

Finally, by (3.2) and (3.3) and the fact that

the proof follows. 0

Next, a lower bound for the rate of convergence of the sequence {In(f)) will be given (compare with Lemma 3.8 in [Lub83]).

LEMMA 3.6. Let f be a function given by (2.4) with nonnegative coeficients, i.e., f; > 0 and f3- 2 0. Then

a) I,(f) < cc if and only if CEO f;cj + CEl f3-c-j < 00

b) If Ia ( f ) < m , then E n ( f ) = I a ( f ) - In ( f ) 2 ~nf;,)+l + PnfGn)+l where m = j," ~ ~ ( x ) ~ x - p ( ~ ~ ) d a ( x ) and PrL = Qn (0)-' lom Q ~ ( x ) ' x - ( P ( ~ ) + ')da(x).


PROOF. a) Since the partial sums in CpO f:xj and C z l f;x-j increase monotonically for x E (0 , co), the proof follows from Lebesgue's monotone convergence theorem [Rud74, Theorem 1.261.

b) B y part a) in this lemma, we can write

On the other hand, from the error expansion for In( f ) (see [BDMGV098])

E , ( x ~ ) ) 0 , 'v'k E Z.

Hence

(3.4) E n ( f ) 2 fpjn)+l ~ , ( ~ - ( p ( n ) + l ) ) + f zn )+l ~ , ( ~ q ( n ) + l 1. Let us denote b y L n ( & x ) the Hermite interpolant from A-,(n),q(n) for the function 4, at the nodes XI,, . . . , x,, of the quadrature formula. B y [BDMGV097b], it follows that

E, ( x k ) = I , ( x k - L, ( x k ; x ) ) .

Taking k = q ( n ) + 1 , then

Therefore,

Now, for k = - ( p ( n ) + I ) , we have

~ , ( ~ - ( p ( " ) + l ) ; x ) = x - ( ~ ( ~ ) + l ) [l - k n Q n ( x ) 2 ]

with kn = Qn(0) -2 > 0 as it can be easily checked. Hence

and this yields

The proof now follows from (3.4), (3.5) and (3.6).

We conclude this section with a number of corollaries of the previous results.

COROLLARY 3.7. Let $(8 ) and l / $ * ( 8 ) be finite. Under the same conditions as i n Lemma 3.1 and Lemma 3.2 and Theorem 3.5, the following holds.

a) If P( f ') < l / q + and p( f -) < l / q - , then

lim I I a ( f ) - ~ ~ ( f ) l l J ~ " = 0. n + x


b) If p(f +) = l /q+ and p( f - ) < l /q-, then

provided that oR( f+ ; &)$J(O) < 1.

c) If p( f +) < 1/q+ and p( f -) = l /q-, then

provided that a,.( f p ; $)/$*(8) < 1. d) If p( f +) = l /qf and p( f -) = l /q-, then

iIa(f) - 1 , ( f ) 1 ~ ' ~ ~ < m a ~ { q + ~ rl-)l (q+, q- as above) n i x

provided that aR(f+; &)$(O) < 1 and ar(f-; $)I$*(@) < 1.

REMARK 3.2. In the above corollary we have assumed that there exist positive constants c+, c-, q+ and q- such that for sufficiently large x: R, 5 (c+z)gt and

r;' 5 (c-x)q-. When in both of these relations equality occurs, then it can be checked that the estimates of the rate of convergence are less than 1.

Now, making use of Lemma 3.6 and definitions (2.14) and (2.17), it follows that

COROLLARY 3.8. Let f be a function as given i n (2.4) with nonnegative coefficients such that I,(f) < m . Then

Finally, we can state

COROLLARY 3.9. Under the same hypotheszs as zn Lemma 3.1, Lemma 3.2, and Corollary 3.8, we have

a) If p ( f + ) > 119: or p(f -1 > l/qF , then

lim IIa(f) - 1~,(f)1'/~'" m . TL-x

b) If p( f + ) = l/q; and p( f - ) = l/q,, then

IIa(f) - ~ n ( f ) l ~ / ~ ' ' > nla~{77:,77F), n-x

4. An application to a family of distributions

The general results of Section 3 will now be applied to a family of weight functions introduced in [LLMF95] and also studied in [BHMF96]. Such weight functions were also considered in our previous papers [BDMGV097a, BDMGV097b, BDMGV0981. Throughout this section, the distribution will be given by

(4.1) da (x ) = A, (x)dx = xu exp{-(xY + X - ~ ) ) ~ X , x E (0, m )


where the parameter a E IR is not relevant for our results. The important parameter is y E (112, m ) . Let {A,) be a sequence of nonnegative integers such that 0 < A, 5 2n and satisfying (2.9). Let us denote by h,,(x,w,) the mth orthonormal

polynomial with respect to x p X n w 7 ( ~ ) d z and by h,,(x, w,) the corresponding mth monic orthogonal polynomial, i.e.,

where h,,,(x, w,) = K,, (wy )xn' + lower degree terms. For our further development, and to make the paper more self-contained, we

recall several results contained in [LLMF95],[BHMF96], concerning the zeros of h,,,(x, w,) which are needed to estimate the rate of convergence.

THEOREM 4.1. i) Let { ~ ~ , , ) 3 n , ~ be the zeros of h,,,,(z,w,). Then. there exzst posztzve constants R,, T, and K > 1 such that z,, E [Tn/K, KR,,] where

1 11-7 + o(n117) and

= [XI + o ( n l ' ~ ) rn ?B(-,)

with B(y) = Lw, 6 T(-r+l) and r ( z ) being the Euler Gamma function. ii) If 0 5 8 < 1 , then

Furthermore, i f 2 = 8 + o(&) as n + oo, then

Set A, = [T,, R,] and (compare with (2.16)

where //PIIz = I P ~ , ( X ) ~ ~ X ~ ~ ~ ~ W ~ ( ~ ) ~ Z . Let @a,, (z) be the conformal transformation mapping the exterior of the unit

circle onto \ A,, , preserving the point at infinity:

with T,, (z) = (z - R, ) (z - T,, ). Moreover ga,, (z; m) and gat, (z; 0) are the Green functions of A,, with pole at infinity and at zero respectively. Note ga,, (z; m ) =

1% PA, , (z)l. For a given Bore1 measure ,LL we let V,(z) = log Iz-tlp1dp(t) be its logarithmic

potential. With this notation. we can formulate:

THEOREM 4.2. Let d(z, A,,) denote the dzstance from z to A,, . Then

4 z , A n ) K f 1 ( z , ~ ? ) < I ~ A , , (z)j exp{w,, - 2V,,, (z)}


where p, is the solution of the following equilibrium potential problem

2vp,, ( 2 ) + x - J + x - ~ + An log x = W n ; x E supp(pn) { > wn; x E it j supp(pn)

with w,, the corresponding equilibrium constant. Furthermore, i t holds V z E C that

en(^; f-,) + 2Vp,, ( 2 ) + [2n - AnIgA, ( 2 ; m) + An [log lzl + gA, ( z ; 0 ) ] = W n ,

where 8,(z; f,) zs the solutzon of the Dzrzchlet problem for the functzon f,(x) =

xY + x P Y on An whzch zs gzven by

These two theorems enable us t o state the following

THEOREM 4.3. Let { P ( ~ ) ) , ~ W be a sequence of nonnegative integers with 0 5 p ( n ) < 2 n - 1. Assume that (2.9) holds. Let I,( f ) be the n-point Gauss-type quadrature formula i n with q ( n ) = 2 n - 1 - p ( n ) for the distribution dcu(x) = wy ( x ) d x , x E [0, co) with w, ( x ) given by (4.1). Then, for any function f satisfying (2.4) it holds that

1-8

n - x

PROOF. By Part i) o f Theorem 4.1, the functions R, and r , as defined in Section 2 can here be taken t o b e ( K > 1 )

(4.4) 1 20

and - r , = ( K ? - ~ ) y B ( 7 ) .

Thus

$(O) = l im - n - x RL

2(1-8)

b y (2.10). O n the other hand

rv* (n ) $*(el = lim - = ( ~ ) l / , = 1. ,L+x r ' ~ L(1-8, n - x v* (n )20

T h e proof now follows b y Theorem 3.5. 0

THEOREM 4.4. Let f be a function as given by (2.4) with nonnegative coeficients. Assume that I,( f ) < co. Then, under the same conditions as i n Theo- rem 4.3, it holds that

with

with K given by (4.4).


PROOF. By Corollary 3.9, we only need lower bounds for p(B) and for ~ ( 8 ) which were defined in (2.14) and (2.17) respectively. We have

l/s(n) - 'Yn cX2 A

p(f3) = lirn --- n+.x. R ~ ( ' ~ 1

with 7, = 1 h:, (x)x-P(")w, ( s ) ~ x .

Thus y, = [~,,(w,)]-~ with K,,(w,) given by (4.2). Now, by (4.4) and Part ii) in Theorem 3.5, it can be easily deduced that

1 ~ ( 0 ) = 4(eK)lly '

On the other hand, by (2.17) and (4.3), one has

Here K,(z, w,) denotes the Christoffel function associated with x-(p(,)+l) wy (5),

while K, (z, w,) represents the one corresponding to x-p(")w, (x). For our purposes, we can use K, (z, w,) instead of K, (z, w,).

Now recall that A, = [?,, R,], hence d(0, A,) = F,. Thus, by Theorem 4.2, it follows

/aAn (0) 1 2 n + 1 - p ( n ) Kn(0,wy) 5 e ~ ( n ) l i m - o (log ~ z ~ + u , ( z : ~ ) ) e ~ n ( ~ . f , )

iTF,

Therefore

It is known that b - a

lirn {g[a.61 (z; m ) - log lzl) = - log Cap [a, b] = - log - Iz/-= 4 '

(log Cap is the logarithmic capacity) and since g ~ , (z; 0) = gA,l (1/z; m), (we denote A;' = {z : l / z E A,})

lirn {ga,, ( l /z ; m ) + log Izl} = - log Cap A;' z-0

which gives - R, - f,

(4.6) lim {gA, (z; 0) + log lzl} = - log -. 2-0 4 R, F,

By the definition of O,(z; f,), we have

where

and it holds that

1 q y + i / 2 ) 1 X, ~ ( y ) := lim. ~ , ( y ) = - = ;l d-dx.

n-cx J;; r'(7 + 1) x(1 - 2)

Therefore

lim & ( O ; f,) - -

-W?) B - n - x

= -B(?)l_H' RZ l i n ~ , ~ - ~ (RnFnlY or equivalently

(4.7) lim B,(O, fY)TI; = -B(?). n - x

Thus, by (4.5) and (4 .7) , we find after some calculations that -

- r&g lim 28 1

2- TL-x &(o, u Y ) l / p ( n ) 4 K e l / r '

This proves the theorem. 0

To end this section, we apply the results for the family of distributions (4.1) to certain integrands f . We consider integrands which can be described as follows. Let

35 X

k=O k=O

be given, with f l ( z ) and f 2 ( l / z ) entire functions. Then consider integrands f ( z ) =

f l ( ~ ) f 2 ( ~ ) , SO that X T

3 =0 j = 1

where

In order to fix the ideas, let us take

Thus

and X

(@)' f: = "' x (21)! (21 + k ) ) !

/ = I

Now by (4 .4) , we can write

2(1 - 0 ) R , = ( K ' x ) " ~ , with K' = K Y -------

?W?) . > I

RATEOFCONVERGENCEFORQUADRATURE

and

where B(y) is given in Theorem 4.1. We can deduce (s > 0)

while

So, by Theorem 4.3, we have for any a, ,6 E R and y > 1,

lirn IIn(f) - ~ ~ ( f ) l ~ l ~ ~ = 0. 7L'X

If y = 1, then one has:

G IIn(f) - ~ ~ ( f ) / l / ~ " < rnax n i x 2(1 - 8)

When considering the so-called balanced situation where p(n) = n for all n (see [JNT83]), then obviously 8 = 112 and we then have

where the last equality holds because of the relation between K , K' and K"

References

[BDMGV097a] A. Bultheel, C. Diaz-Mendoza, P. Gonzblez-Vera, and R. Orive, Quadrature on the half lzne and two-poznt Pad6 approximants to Stieltjes functzons. Part II: Convergence, J . Comput. Appl. Math. 77 (1997), 53-76.

IBDMGVO97bI A. Bultheel, C. Diaz-Mendoza, P. Gonzalez-Vera, and R. Orive, Quadrature on the half line and two-point Pad6 approximants to Stzeltjes functzons. Part I I I : The unbounded case, J . Comput. Appl. Math. 87 (1997), 95-117. A. Bultheel, C. Diaz-Mendoza, P. Gonzilez-Vera, and R. Orive, O n the convergence of certazn Gauss-type quadrature fonnulas for unbounded intervals, Math. Comp. (1998), Accepted. M. Bello-Hernandez and A. Martinez-Finkelshtein, Zero asymptotics of Laurent orthogonal polynomials, J . Approx. Theory 85 (1996), 324 342. W.B. Jones, 0 . Njbstad, and W.J . Thron, Two-poznt Pad6 expanszons for a family of analytic functzons, J . Comput. Appl. Math. 9 (1983), 105124. G. L6pez-Lagomasino and A. Martinez-Finkelshtein, Rate of convergence of two- point Pad6 approxzmants and logarithmic asymptotics of Laurent-type orthogonal polynomzals, Constr. Approx. 11 (1995), 255-286. D. Lubinsky, Geometric convergence of Lagrangzan interpolatzon and numerzcal zntegration rules over unbounded contours and zntervals, J . Approx. Theory 39 (1983), 338--360. H.N. Mhaskar, Wezghted polynomial approxzmatzon, World Scientific, 1996. W . Rudin, Real and complex analysis, 2nd ed., McGraw-Hill, New York, 1974.

DEPARTMENT OF COMPUTER SCIENCE, K.U.LEUVEN, LEUVEN, BELGIUM E-mail address: Adhemar. BultheelOcs . kuleuven. ac . be URL:http://www.cs.kuleuven.ac.be/-ade/

DEPARTMENT ANALISIS MATH., UNIV. LA LAGUNA, TENERIFE, SPAIN E-mail address: cjdiazQul1. es

DEPARTMENT ANAIJSIS MATH., UNIV. LA LAGUNA, TENERIFE, SPAIN E-mail address: pglezQul1 . es

DEPARTMENT ANALISIS MATH., UNIV. LA LAGUNA, TENERIFE, SPAIN E-mail address: rorive(0ull. es


Wavelets by orthogonal rational kernels

Adhemar Bultheel and Pablo Gonzblez-Vera

Dedicated to Jerry Lunge on the occasion of his 70th birthday

ABSTRACT. Let L, be the space of polynomials or rational functions of degree n at most with poles in a prescribed set of numbers inside the unit disk. Consider a complex inner product with respect to a finite positive measure on the unit circle. Suppose {$k);=o is the corresponding orthonornal basis for 13,. - Set n = 2, and V, = L,. Then k, ( z , w ) = C;=,, $k (z)$k ( w ) , is a reproducing kernel for V,. For fixed w , such reproducing kernels are known to be functions localized in the neighborhood of z = w . Moreover, by an appropriate choice of the parameters {Enk);=O, the functions {cp,,k(z) = kn(z, E,k));=, will be an orthogonal basis for V,. The orthogonal complement W , = V,+1 8 V, is spanned by the functions {$~ , ,k ( z ) -- 1,(z, q,k));=: for an appropriate choice of the parameters { l l n k ) ; ~ ~ where I, = kz, - k, is the reproducing kernel for W,. These observations form the basic ingredients for the construction of rational wavelets on the unit circle with respect to an arbitrary positive measure.

1. Introduction

The main motivation for this paper is the basic observation that was made by Fischer and Prestin [FP97] in the case of polynomials on the real line.

This observation is that a reproducing kernel in a reproducing kernel Hilbert space has the property that it is a localized function. If 'H is a Hilbert space of functions defined on some set X c @, then it is called a reproducing kernel Hilbert space if there is a kernel function k , E 'H (w E X is a parameter) such that (f, k,) = f(w) for all f E 31,w E X. It has the remarkable property that it gives the solution of infpEX{ll f 11 : f (w) = 1, w E X) namely f = k w / k w (w). This can be interpreted as follows: without the constraint f (w) = 1, the solution would be f = 0. The constraint forces f to take the value 1 at w, but the optimization problem implies that in X \ {w), the function f is "as small as possible". If the

1991 Mathematics Subject Classification. Primary 42C05; Secondary 46322, 42A38. The work of the first author was performed as part of the project "Orthogonal systems and

their applications" of the FWO under grant #G0278.97. The work of the second author was supported by the scientific research project of the Spanish

D.G.I.C.Y.T. under contract PB96-1029.


102 ADHEMAR BULTHEEL AND PABLO GONZALEZ-VERA

space 'H consists of "smooth" functions like polynomials or rationals, then f has typically a "bump" in z = w and much smaller oscillations away from z = w.

Fischer and Prestin exploited this fact to construct polynomial wavelets. Con- sider the space of polynomials on an interval (possibly infinite) [a, b] & R equiped

b with the inner product (p, q) = Ja p(x)q(x)dp(x) where p is a finite positive measure. The subspaces C, of polynomials of degree at most n form a nested sequence Lo C C1 C . . . C C, c . . . of subspaces. If f, is the projection of some f E L2(p, EX) onto C,, it is clear that for increasing n we shall get better and better approximations for f , i.e., more and more details of f will become visible in the projections f,. One may say that the C, represent different scales or resolution levels of the space LZ(P, EX).

More generally, in multiresolution analysis, one decomposes a function space in several resolution levels and the idea is to represent the functions from that function space by a low resolution approximation and adding to it the successive details that lift it to resolution levels of increasing detail.

Wavelet analysis couples the multiresolution idea with a special choice of bases for the different resolution spaces and for the wavelet spaces that represent the difference between successive resolution spaces. Thus if V, are the resolution spaces: Vo c V1 c . . . , then the wavelet spaces are given by the differences: W, = VS+l eV,.

By their definition, the functions in W, will automatically be band-limited in their detail information. This means that the basis functions for these W, (which are called wavelet functions) will have Fourier transforms that mainly live on a relatively small interval i.e., they are small outside that interval. Moreover, for true wavelet functions, one wants the functions themselves to live mainly on a small interval as well. Then a local change in the function f will only have considerable effect on the components along the basis functions that have their main part in the neighborhood of the change and it will much less affect the other components. Conversely, changing one component will only have a local effect on the function.

The kernel functions which we described above do have this wavelet-like behavior and the problem is thus whether it is possible to give a basis of reproducing kernels that span the successive resolution spaces and wavelet spaces. The answer to this question is affirmative and is completely analyzed in [FP97] for L2(p, R) where the subspaces are spanned by polynomials.

In this paper we give two modifications of this idea. First, we consider complex function spaces where orthogonality is with respect to a measure on the complex unit circle and in the second place we generalize the space of polynomials to the space of rational functions whose poles are in a certain prescribed set. To be more precise, C, will be the space of functions of the form p , /~ , where p, is a polynomial of degree at most n and .rr,(z) = n:=l(l - h,z). We assume that all the cyk are inside the open unit disk. A parallel theory could have been developed where all the a k are on the unit circle, but we shall not do this here. A similar discussion for rational wavelets on the real line with all the poles on the real line was given in [BGV98].

The results presented in this paper rely on two pilars: the theory of wavelets, which has to be generalized at different places, very much like it was done in [FP97] and the theory of orthogonal rational functions on the unit circle.

The plan of the paper js as follows. First we introduce the rational function spaces we shall work with and give the basic ideas of multiresolution analysis. In

WAVELETS BY ORTHOGONAL RATIONAL KERNELS 103

Section 4 we summarize the main properties of reproducing kernels and establish a way to make them produce a basis. In the next two sections we discuss orthogonal basis functions. First the basis of orthogonal rational functions (ORF) and in the next section it is shown how one can find a basis of orthogonal reproducing kernels (ORK). In Section 7 gives a discussion of how to construct a wavelet basis of reproducing kernels (WRK), but it is not obvious how this can be made orthogonal in a general case. Section 8 describes the dual (i.e., biorthogonal) basis functions and this is used in the next section to give the explicit formulas for analysis and synthesis computations. Next, the generalization of multiresolution analysis (MRA) and completeness properties are given and some theorems about the symmetry of the wavelets and scaling functions. Finally, in Section 12, some computational aspects are discussed.

2. The rational functions

We consider the complex Hilbert space L2 = L2(T1 p) of functions which are square integrable on the unit circle T = { z E @ : lzl = 1) with respect to some finite positive measure p. The inner product is given by

For simplicity we assume that ST d p = 1. Denote by II, the set of polynomials of degree at most n, then for a given sequence of points a l , a2 , . . . all inside the unit disk D = {z E C : Izl < l), we consider the spaces of rational functions

where TO = 1 and T, = JJ;=,(l - Ekz), n = 1,2 , . . . . We refer to the "polynomial case" as corresponding to the choice crk = 0 for all k = 1,2 , . . . . In the polynomial case, all the poles are at infinity and we have II, = C,.

An obvious choice to represent these spaces is to construct an orthonormal basis, i.e., a sequence of functions (60 , 61, . . . ) such that 4, E C, \C,- 1, 4, I C,- 1

and Ilq5,ll = 1. Such functions were first considered by M.M. Djrbashian in the sixties (see the summary [DjrSO]) and later by A. Bultheel, K. Pan, X. Li, and in a series of papers by E. Hendriksen and 0. Njdstad in collaboration with the authors. Most of the results we need are found in one of these papers, but we do not reference all of them. Instead we refer to a forthcoming monograph [BGVHN98b] which brings together all these results in a systematic way, or to a preliminary report [BGVHNSO] which collects many of the properties needed.

3. Wavelet transform: decomposition and reconstruction

As explained in the introduction, we consider a multiresolution structure defined by nested subspaces C, for increasing n. More precisely, we consider the spaces V, = Cz. for s = 0,1 , . . . . Thus the resolution spaces V, have dimension 2, i 1, s > 0. We define the wavelet spaces W, as the orthogonal complement of V, in V,+l, thus W, = V,+l @V,. The dimension of W, is 2,. To simplify the notation we shall henceforth assume that n = 2' and work with the resolution spaces C, and the wavelet spaces K, = La, @ C,. To get started, we need the initial spaces V- 1 = Lo and W- = KO = C1 8 Lo. Both have dimension 1.

REMARK 3.1. We could avoid this special treatment of the initial spaces by defining Vs = L2.-1, and Ws = Vs+l 8 Vs for s = 0,1 , . . . . Then V3 and Ws both have dimension 2'. However, since the previously mentioned choice was made in [FP97], we keep the same kind of convention in this text.

The projection fn of f E L2 onto Ln = Vs is called the representation of f at scale or resolution level s. For every scale s, we can write fin = fn + gn where f2n 6 L2n, f n E Cn, and gn E K,. The projection fn is the low resolution part and gn is the high resolution part of fin. The low resolution part is decomposed again as fn = fn12 + gn/2 etc. Thus, after a number of such decompositions we find

Thus

or equivalently

Now suppose that for n = Y , s = 0,1, . . . , the wavelet spaces Kn are spanned by localized basis functions (wavelet functions) {$n,k : k = 0 , . . . , n - 1) and the resolution spaces Ln are spanned by localized basis functions (scaling functions) { ( P ~ , ~ : k =0 ,1 , . . . ,n ) .

If We Suppose that fn = pn,k(pn,k and gn = ~ 2 1 ; qn,k$n,k, then we can write

s 2&-1

The set of coefficients { P ~ , ~ ) U {q2k-l,l : k = 0 , . . . , S; 1 = 0, . . . , 2k - 1) is called the (discrete) wavelet transform of fin.

Let us introduce the notation for any integer n 2 0

and

(the superscript means transpose) so that fn = p n q n and gn = qn@,. In one step of the analysis or decomposition algorithm one starts from the coefficients p2, and transforms this in pn and q,, which corresponds to a change of basis like

or more compactly: f2n = p2np2n = pnpn +qn@,. In the next step one transforms pn into pn12 and qn/2 etc. The synthesis or reconstruction algorithm goes in the opposite direction and generates p2n from pn and q,. Thus it is important that we have an efficient algorithm to do the transformations back and forth between

P2n ++ { ~ n 7 9n ). Such a transformation should preferably be unitary for the ease of inversion

and for the numerical stability. Thus the basis {pan$ : k = 0, . . . ,272) as well as


the basis { p n , k : k = 0 , . . . , n ) U {$J,,~ : k = 0 , . . . , n - 1) should be orthogonal. By construction, the &,I, are orthogonal to the p,,l and hence to all p,,~ and $,J with m < n.

An obvious choice for the basis functions would be the orthogonal rational functions (ORF) 4, that we have defined at the end of Section 2. Thus p,,k = 4k for k = 0, . . . , n and $Jn,k = for k = 0 , . . . , n - 1. However, except for very special cases, these functions are not localized and therefore are not wavelet-like. Because reproducing kernels do have this property, we shall try to introduce a basis of orthogonal rational kernel (ORK) functions.

4. Reproducing kernels

The following properties concerning reproducing kernels are well known (see for example [AroFiO, Mes62, Don741)

THEOREM 4.1. W e consider an m-dimensional subspace K of Cn . Assume that its reproducing kernel is denoted as k(z, w). Then

1. For any orthonormal basis {el, e2, . . . , e m ) of K, the reproducing kernel is given by

m

k(z, w) = ek(z)ek(w). k=l

2. For any set of distinct points {wl, w2,. . . , w,) among the points of analyticity for K,

i s positive semi-definite. 3. The orthogonal projection o f f E 13, onto K is given by

P K ~ = ( f ( z ) , k(z, w)) . 4. For any point w where i t makes sense,

inf ( 1 1 f 1 1 ' : f (w) = I) = [k(w, w)]-' f € K

and the minimizer is f (2) = k(z, w)/k(w, w)

Also the following lemma is easy to verify.

LEMMA 4.2. Let {40,. . .4,) be the orthonormal basis for C, as introduced above, then for any set of distinct points x = {xo ,x l , . . . ,x,) which are points of analyticity for functions i n C,, the matrix

~ O ( X O ) . . . 40(xn)

(4.1) @n(x)

4n(50) . . . 4n(xn)

i s regular. I

PROOF. This is obvious because the 4k form a Haar system in the polynomial as well as in the rational case. See Davis [Dav63, Chap. 11, Sect. 2.41. 0

FIGURE 1. The real (left) and imaginary part (right) of the functions kn(z, w) = C:=o(z /~)k (top) and ln(z,w) = C ~ , + ~ ( Z / W ) ~ (bottom) for z = eie and w = 1, plotted as functions of 8. Note that the real parts are even functions, while the imaginary parts are odd functions.

An immediate consequence is that we can choose a basis of reproducing kernels for the spaces C,.

COROLLARY 4.3. If k,(z, w) i s the reproducing kernel for 13, and the x =

{ x ~ ) ; = ~ are n + 1 distinct points of analyticity for Ln, then the functions {ynj (z) = kn(z, x~)):=~ form a basis for 13,.

PROOF. Since, with the matrix @,(x) as in the previous lemma and denoting by a superscript the complex conjugate transpose:

the statement follows because {+k);=o is a basis and @,(x) is regular. 0

EXAMPLE 4.1. Let us give a very simple example. Consider the polynomial case, i.e., all a k = 0 and assume that the measure p is the normalized Lebesgue measure on T. Then the orthonormal polynomials are just dk(z) = zk. This is the basis used in classical Fourier analysis. Each basis function has a Fourier transform which is a Dirac impulse and is thus extremely localized, but the functions themselves are in modulus equal to 1 everywhere on T and thus they have the worst possible localization on T. However the kernels kzn (z, w) = CEO zkmk are

n

, 2 = eie, , = eiw. sin $ ( 8 - w) k=-n

Hence these kernels are complex exponentials modulated by the Dirichlet kernel, which is localized around 6 = w. This can be verified in Figure 1. For the spaces Kn = Can 8 C,, the reproducing kernels are given by l,(z, w) = kzn(z, w) - kn(z, W) = ~ ~ , + , +k(z)+k(w). In our example this is just (zE)"+' C;:: The plots for this trivial example are given in the bottom rows of Figure 1. It


is immediately observed that these oscillate more but otherwise have properties similar to the kernels for C,.

Although we have plotted the kernel I, for the space K, in the previous example, it is not as trivial as for the k, to conclude that there are numbers {yo,. . . , yn-1) such that {l,(z, Yk)) ;~ i will form a basis for K,. Indeed, the system {+k)gn+l will not be a Haar system in general. Before we investigate this any further, we shall first recall several properties of the ORF basis {dk) in the next section.

5. The ORF basis

We need several properties of the orthogonal rational functions (ORF) 4, that were defined in Section 2. Almost all of them can be found in [BGVHN98b]. We shall only give a proof of those properties that are new.

We start by giving some notation. Besides the points {al, a 2 , . . . ) c !ID, we define the special point CYO = 0. Furthermore, we define for k = 1 , 2 , . . . the Blaschke factors

- z - a k a k &(z) = Zk --- , with zk = 1 if a k = 0 and zk = -- otherwise,

1 - 5k.z lak 1 and the Blaschke products

For any function f we set f,(z) = f ( l / F ) and if f, E L, (it should be clear from the context what space L, is involved) we define f; (z) = B,(z) f. (z).

Obviously L, = span{Bo, B1,. . . , B,). If we write 4, = anoBo +anlB1 +. . .+ a,,B,, then the orthonormal 4, are uniquely defined if we impose the normalization n, = +;(a,) > 0. (Note that n, is the a,, from the previous expansion.) We shall use the notation 4, to mean this uniquely defined orthonormal rational function.

The ORF satisfy a recurrence relation that generalizes the Szego recurrence relation for orthogonal polynomials on the unit circle. In fact the polynomials form a special case obtained by setting all a k = 0.

THEOREM 5.1. For the orthonormal basis functions 41, = ~ k B k + . . . , Kk > 0 in C,, a recursion of the following form exists

and its superstar conjugate is

The parameter A, is given by

where O 5 k I n - 1.


The constant en > 0 defines the modulus and i t is given as the positive square root of

(Note IXnI < 1.) The vnl and vn2 are constants on T which arrange for K, = $:(an) > 0. They

are related by 7 7 , ~ = znqnl. The initial conditions are q50 = 4: = 1.

For the development of the theory of rational wavelets that we give in the next sections, the precise recursion is irrelevant. However, when performing the proper computations, the availability of a recurrence relation is really important, since a general Gram-Schmidt procedure to compute the ORF basis is unacceptable, because of all the inner products (i.e., integrals) to be evaluated.

In this perspective it is important to have good recurrence relations for the ORF basis, or even for the reproducing kernels (see later). The above recurrence is not the only one that one could use. We give some samples of theoretically equivalent recursions, which may have a quite different numerical behaviour. These recurrences are all related to continued fractions.

For example, the previous recurrences are related to a continued fraction of the form

where

Such a continued fraction is (equivalent to) a Perron-Carathkodory (or PC) fraction when all a k = 0 and it is called a Nevanlinna-Pick (or NP) fraction in the general case because it is related to the solution of the Nevanlinna-Pick interpolation problem (see Section 12). The successive convergents are

where the $k are the ORF and the $k are the so called associated functions or functions of the second kind.

The successive approximants are interpolating functions for the Riesz-Herglotz transform of the measure:

in the sense that

with h(z) analytic in Izl > 1 while h(z) is analytic in lzl < 1. Several variants of this continued fraction can be obtained for example by

applying several equivalence transforms or contractions. Alternatively, they may be


derived from the recurrence relations obtained by eliminating some terms from the coupled recursions (5.1-5.2) given in the previous theorem. For example, one could eliminate c$i - l from (5.1) which would give a relation between c$,, 4: and q5n-l. We shall not mention all of them. Several possibilities are given in [BGVHN98b].

The most prominent one of the alternatives is some variant of a three-term recurrence relation, which can be written in the form

will give f k = c$k. The other solution is given by functions of the second kind. In the polynomial case, the continued fraction corresponding to this recurrence

is related to an M-fraction and the corresponding recurrence for the q5: is related to T-fractions. The convergents of these continued fractions are related to two-point Pad6 approximants. In the general case these become multipoint Pad4 approximants and the continued fractions are sometimes called MP-fractions. Many more of these continued fractions related to ORF are described in [BGVHNgga].

As we have seen, it is not exactly the ORF basis that we want to use, but the basis of reproducing kernels. Of course, once we know the ORF basis, we can compute the reproducing kernel basis, but it would be much more interesting if we had a recurrence directly for the reproducing kernels. Such a recurrence does indeed exist. In fact it is even simpler to formulate.

THEOREM 5.2. Let kn(z, w) be the reproducing kernels for the spaces C,. Then

and

where

The superstar is with respect to the variable z.

Note that the pn(w) of this theorem and the An of the previous theorem are related by

However, the recurrence for the kernels only holds for z, w @ T. For w E T, it is clear that pn(w) E T, so that 1 - l p , ( ~ ) ( ~ = 0, and the recurrence breaks down. In the sections to follow, we do need the kernels for values of w E T, so that the recurrence of the previous theorem is useless.

Like in the polynomial case we have a certain equivalence between the existence of a recurrence relation and the orthogonality (Favard's theorem) [BGVHN92] and

on the other hand between the recurrence relation and the existence of a Christoffel- Darboux type formula (see [BreSO] for the real line). In our case the Christoffel- Darboux relation takes the following form.

THEOREM 5.3 (Christoffel-Darboux formula). Let {+k)c=o with E Ck\Ck-1 be an orthonormal basis for Cn, then the reproducing kernel kn(z, w) for C, satisfies

For this paper, the relation with quadrature formulas will be important. The zeros of the orthogonal rational functions (ORF) are known to be in the open unit disk Dl so they are not suitable for the construction of the quadrature formulas representing integrals over the unit circle T. We should rather have knots which are located on T. Such knots are provided by the zeros of para-orthogonal functions. The para-orthogonal functions are defined as

Qn(z1rn) = &(z) +rn+:(z)

for rn E T, n = 0,1, . . . and the zeros of these are simple and on T . We have the following rational Szego quadrature formula.

THEOREM 5.4 (Rational Szego quadrature). The zeros of the para-orthogonal rational function Qn+l(z, rnf l ) are all simple and on T . Let us denote them by {<n,k)c=o. Moreover, defining Xnk = [k, (Cnk , Cnk)] -' , with k,(z, w) the reproducing kernel for C,, then equality holds i n

n,

( f i g ) = ~ n k f (~nk)~*( tnk ) l v f 1 E Cn. k=O

Conversely, i f the above equality holds, then the Erik should be the zeros of the para- orthogonal function Q,+l(z, rn+1) for some rn+l E T and Xnk = [k,(Cnk, tTLk)]-l.

The n + 1 zeros {<nk);=O can also be characterized as follows.

THEOREM 5.5. Let kn(z, w) be the reproducing kernel for C,. Define Cno = w with w arbitrary on T and {&k);=l the n zeros of k,(z, w). Then the numbers {Jnk);=O are the zeros of the para-orthogonal function Q,+l (z, T,+~) with

Conversely, if {Jnk);=O are the zeros of some para-orthogonal function Qn+l(zl rn+l), then there exists a number w E T , such that Cno = w and r n + ~ = -w, while {<nk};=l a n the n zeros of k,(z, w).

n+l

PROOF. First note that neither nor can be zero on T. Thus rn+l is well defined and is on T for all w E T because I+:+, 1 = I Bnfl 1 1 = 1 on T. Thus it is immediately seen that the zeros of Qn+1(z, r,+l) are the zeros of

Obviously z = w is one of the zeros. In the Christoffel-Darboux formula, this zero is canceled by the denominator and it is of course not a zero of k,(z, w). The n other zeros of Qn+l(z, T ~ + ~ ) however are precisely the zeros of kn(z, w).

Conversely, recall that all the zeros of +,+I are in D and hence none of the zeros of are in D U T. Thus it follows that has winding number


n+ 1 , SO that there are n + 1 values of w E 'IT such that T,+I = - & + ~ ( w ) / ~ ~ + , ( w ) . Each of these values of w has to be a zero of Q n + l ( z 3 ~ , , + 1 ) : thus we can pick the one which is En@ The other zeros of Q,+l(z, ~ , + 1 ) have to be zeros of k n ( z , w) as was explained above. 0

6. The ORK basis

We have discussed before why we want to replace the ORF basis by a basis formed by reproducing kernels. By Corollary 4.3. we know that { k , ( z . ~ , ) } ~ = ~ forms a basis for Cn for almost any set of distinct points x = {x,}. The question is whether it is possible to choose the points in x such that this basis is orthogonal In that case we would have a basis of orthogonal rational kernels (ORK).

It turns out that if we choose x, = <,,. j = 0 , . . . .n the zeros of a para- orthogonal function Q n + l ( z , T,,+I), then the basis {,-,,(z) = k , (~ .<, , ) ) ;=~ is orthogonal.

THEOREM 6.1. Let k n ( z , w) be the reproducing kernel for C,, and let En =

{[,,)y==O be the zeros of a para-orthogonal function Q n + l ( z . for some E T. Then the basis for C , defined by

p n , ( z ) = k , ( z , < n , ) ~ j = O , l , . . . , n

is orthogonal.

PROOF. Let @ , = @,(En) be defined by (4.1) then it follows from (4 .2) and from

kn(Enj,Inz) = 6zjkn(Eni: En%)) i : j = 0 : . . . : n.

In terms of the matrix (4.1) this reads

@ @ = A or equivalently @,An@: = I

with An = diag(Ano,. . . , Ann) a diagonal matrix with An, = l/k,(<,,. t,,). ViTriting @,An@: = I explicitly gives

n

Ank iZ(~nk) '$ j (<nk) 623, % : 3 -- 0 , . . . k=O

Because also (4,: 4,) = 6,, for 2, 3 = 0 , . . . . n, this means that the above quadrature formula is exact for the inner product of all basis functions in C,. hence for the inner prcduct of any two functions in 13,. By Theorem 5.4. this means that it can only be the rational Szego quadrature formula. Thus the theorem follows. 0

E X A M P L E 6.1. Let us consider again the trivial example where all a k = 0 and where p is the normalized Lebesgue measure on T. It is clear that if we set T,+I =

-1: then Q,+I(z, T ~ + ' ) = zn+'-1. Thus its zeros are given by {<,,k = e-"k"/(n+l) } for k = 0 , . . . ; n. Supposing that f , E 13, is given by f n ( z ) = C2=oankq5k(z) =

C y = o ~ n j k . n ( z r En?): thus


it follows that the {ank) are given by the Discrete Fourier Transform (DFT) of {p,,). In other words, the matrix @,(En) becomes a Fourier matrix.

7. The WRK basis

We now consider the problem to find a reproducing kernel basis for the spaces K, = Lz, 8 L,. Obviously the reproducing kernel for K, is

2n

L ( z , W) = kan(z3 W) - k ( z > W) = C Q ~ ( z ) ~ I ; ( w ) .

The main question in this respect is: Can we find n numbers {rl,j),"zi such that the functions {$,j(z) = l , ( ~ , ~ , ~ ) ) , " ~ i form a basis for K, and if possible, can it be made orthogonal?

We have t o investigate the matrix

I + l o . . . 4n+l(~,-1)

(7.1) * n ( ~ ) =

42, (YO) . . . Q2n(~n-1)

points {1J3),"zi on T which make this matrix regular.

1 Using the following lemma, it is possible to prove that there always exists a set of

LEMMA 7.1. Assume that Q E C(2n+1)x(2nf1) is a square matrix such that QHQ = D with D regular and diagonal. Assume moreover that this Q i s subdivided as

with Q1 E @(n+l )x (n+ i ) and hence Q4 E Cnxn . If Ql i s regular and i f Q2 E enxn+l is of full rank n, then Qq is regular.

PROOF. Let D be subdivided into the two parts Dl E and Dz E ~ n x n . We know from QHQ = D that

Q ~ Q ~ + QFQ, = o so that rank(QFQ1) = rank(QFQ3). Because Q1 is regular and rank(Q2) = n , we find that rank(QFQ3) = n. Now if Q4 were singular, then there would exist a nonzero vector c E @ I X n such that c Q ~ = 0, hence also cQFQ3 = 0. In other words, rank(QrQ3) < n, which is a contradiction. 0

This makes it possible to prove the following theorem which ensures the existence of a wavelet reproducing kernel basis (WRK).

THEOREM 7.2. Consider the zeros Ezn = {<z, ,~ : k = 0, 1, . . . ,212) of the para- orthogonal function Q~,+ l ( z , r2,+l) for some Tzn+l € T. If we select y , = {yk : k = 0 , . . . , n - 1) to be any n out of the 2n+1 zeros i n J2,, then the matrix Q,(y,) as defined i n (7.1) will be regular.

PROOF. Since the ordering of the zeros is completely arbitrary, we can always assume that we select the yk to be the last elements of E2, = : J = 0 , . . . ,2n) . Row consider the matrix Q2, = @2n(<2n) of (4.1) where the evaluation is in the points of tan, then +,(y,) appears as the n x n right lower part of the matrix @an. By Lemma 4.2. it follows that taking the block of the first n + 1


columns in QZn, any selection of k different rows from it will result in a matrix of rank min{k, n + 1). Thus the conditions of the previous lemma are satisfied and the theorem follows. 0

This theorem settles the question of the existence of a WRK basis of the form { ln (z lYk)}~ ,~ . It is however not clear how to choose these { Y k ) ~ ~ ~ to make the WRK basis orthogonal, if it is possible at all. The same problem was encountered for the polynomial case on an interval in [FP97] where an orthogonal WRK basis could only be constructed for a Chebyshev weight of the second kind.

EXAMPLE 7.1. For some particular cases, it is however possible to construct such an orthogonal WRK basis. We give a trivial example. Consider again the case where all cur, = 0 and where orthogonality is with respect to the normalized Lebesgue measure of T. In that case

Note that 13, = lIn and K, = C2, @ Cn = Z ~ + ' I I , - ~ Furthermore, for some distinct qnk, k = 0 , . . . , n - 1, all on T define the functions gnk(z) = l,(z, qnk), k = 0 , . . . , n - 1. In analogy with Theorem 6.1, for these to be an orthogonal basis of K,, we need the existence of numbers X n k , k = 0 , . . . , n - 1 such that

In other words, we need a rational Szego quadrature formula which is exact in - - zn+l ITn-l. Because f (z) = zn+lpn-l (z) and g(z) = zn+lqn-l (z) are both in

ICn if pn-1, qn-1 E IIn-l, we have

The quadrature formula will be exact if and only if the nodes qnk are the zeros of the para-orthogonal polynomial Qn(z, T ) = zn + T , T E T and the weights are given by

1 1 1

Thus in this case, one can take for example the nth roots of unity for qnk and the basis $nk will be an orthogonal WRK basis. Note that here also the matrix Q,(qn) will be the matrix of a DFT.

We can push this example a bit further in two directions. First by considering orthogonal polynomials with respect to a rational modification of the Lebesgue measure. A rational modification of the Lebesgue measure is defined as a measure p satisfying for some a3 E ID, j = 1, . . . , m

where X is the normalized Lebesgue measure. Orthogonal polynomials for such a measure and corresponding quadrature formulas were considered in [GVSLN96], see also [Sze67, p.289-2901. The orthogonal polynomials for such a measure are given by $k(z) = ~ ~ - ~ h * ( z ) = zkh*(z) for k > m (see [Sze67, Thm. 11.2, p.2891).


For the polynomial h(z) of degree m we have used h* (z) = zmh, (z) = nyzl ((z a,) . Thus for n + l > m, it follows that K, = span{&+l,.. . , 42,) = zn+lWmh* ( z ) K - I . Therefore, i f f , g E IC,, then they can be represented as f (z) = zn+'-'"h* ( z ) ~ n - l ( z ) and g(z) = ~ , + ' - ~ h * ( z )qn-l (z) with p,-1: q,-1 E I I - 1 and we have (f ,g), =

(pn-1: qn-l)X. If we want {$,k(z) = ln(z ,vnk)) to be an orthogonal WRK basis, then, as

shown before, the numbers qnk should be chosen such that there exist positive numbers A,,+ such that (7.2) holds, hence such that

This is clearly possible by choosing qnk t o be the n zeros of the para-orthogonal polynomial zn + T , T E 'F (and the weights are A,,, = lh(q,k)12/n).

A second generalization of the simple case that was discussed above is t o consider orthogonal rational functions with respect to the Lebesgue measure. Such a situation was considered in [BGVHN94] . In this case the orthogonal rational functions are given by

Therefore K, = C Z ~ 8 C, = span{&+l,. . . , $2,) is given by

where CnPl is the space c.f rational functions associated with the points {G, =

C Y , + ~ + , ) ~ ~ : . Thus, if f , g E K,, then there exist E such that

and therefore

with dp(t) = Jt - ~ , + ~ ) - ~ d X ( t ) . Note that this is a rational modification of the Lebesgue measure which is, up to the factor 1 - an+1 l 2 equal to the Poisson kernel P ( ~ , Q , + ~ ) = (1 - / ~ , + ~ ~ ) / l t - a,+1I2. In [BGVHN94 , Section 31 weights and abscisses were given for this case which give an n-point rational Szego quadrature formula which is exact in C(,-l), . hence for which the inner product (7.3) is evaluated exactly. Thus also in this case we can construct an orthogonal wavelet

n- 1 basis { $ n k ( z ) = l n ( ~ , ~ n k ) ) k = ~ .

EXAMPLE 7.2. Let us give a less trivial example. We choose equispaced C Y ~ on the circle izl = 0.9 as shown in Figure 2. The reproducing kernels for the Lebesgue measure on the unit circle are computed and their real, and imaginary part, and their modulus are plotted, both for the kernels k, and for t,he kernels 1,. The result is shown in Figure 2.

WAVELETS BY ORTHOGONAL RATIONAL KERNELS

FIGURE 2. The real and imaginary part and the nlodulus of the functions ks(z, w) (top) and 18(z, w)(bottom) for z = ele and w =

1, plotted as functions of 0.The measure is the normalized Lebesgue measure and the C Y ~ are chosen as a k = 0 . 9 e ~ ~ " ' ~ for k = 1, . . . , 8 and ak+s = 0.9e2(2k+1)a/s for k = 1, . . . ,8. These ak ' s are plotted in the figure on the right.

8. Biorthogonal basis functions and interpolation

For several reasons, it is important to have a biorthogonal basis in wavelet theory. Like in the polynomial case, the biorthogonal basis functions for the reproducing kernel bases are given by interpolation functions. This is what we shall explain in this section.

Let us define the Lagrange polynomials in lT, for the interpolation points x, = {x~) :=~ (all distinct and on T) by

and define

with as before .rr,(z) = nLO(l - 5 , ~ ) . Then obviously Lnk(x,) = dk3 , k , ~ = 0, . . . , n while L,k E C,. We call these Lnk the fundamental Lagrange interpolating functions (FLIF) of C, for the points x, = {x~)"$?~. It immediately follows that for any function f E C, we may write

n

f (2) = C f (xk)Lnk(z!. k=O

Defining in C, the discrete inner product

(f , g),,, = f ( ~ k ) m .


it is directly seen that the FLIF {Lnk)t=O form an orthonormal basis for L, with respect to this inner product. Note that the reproducing kernels for L, with respect to this inner product are given by xy=o Lnj (z)Lnj (xk) = Lnk (z). Thus the L,I, are reproducing kernels and by the general theory we may say that among all functions in C, that take the value 1 at xk, they will have minimal norm IILnk llm,, .

If we set pnk(z) = k , ( ~ , xk ) and gnk(z) = Lnk( i ) , then because of the reproducing property of cp,k

In other words, cpnk and Pnk are biorthogonal bases for L,. Note that we can characterize gnk in another way. We may write gnk =

xy=o cjk)q5, where dk) = [cc) . . . cik)] is defined by dk)an = ek where ex =

10.. . 0 1 0 . . .O] is the kth unit vector and a, = @,(x,) is the matrix (4.1). Thus dk) is row k in the inverse of a,, so that we have

It is clear that if the xk are the zeros of a para-orthogonal polynomial Qn+l(z, ~ n + l ) , then the cp,k are orthogonal and (cp,,, p n j ) = S,jpn,(xi). In that case we have of course Lnk(z) = pnk(Z)/(~nk(xk).

Similarly, given y, = { Y k ) ~ ~ ~ , one can construct a biorthogonal basis for Qnrc(z) = ln ( i , yk), on condition that the matrix Qn = 9,(yn) of (7.1) is regular. Indeed, let e k = [O . . . 0 1 0 . . . O ] be t,he kth unit vector, then if 9, is regular, there

(k) is exactly one solution d(" = [d,,, . . . dg)] to the equation d(k)Q, = e k ? for each ( k ) k = 0, . . . , n - 1. The function Gnk (2) = g,+, d3 Qnj ( i ) is obviously in Kn and

we have qnk(y3) = hk3, k, j = 0 , . . . , n - 1. Thus these Gnk form the FLIF of ICn for the interpolation points y, = { y k ) ~ ~ ~ . We can write for any function f E IC, that

n-1

f (i) = C f (~k)?il'n/c(z). k=O

The 4,k are orthonormal with respect to the discrete inner product

and $,k is the solution to the problem

min{llf lli,, f ( ~ k ) = 1; f E xn). Moreover, they form a biorthogonal basis for the Qnk in IC, because by the repro-

ducing property of Qnk we have ($,*, +nj) = Sxj, P , j = 0 , . . , n - 1. The relation

between the bases

(8.2)


with \kn = \kn(yn).

9. Decomposition and reconstruction formulas

We are now in a position to give concrete formulas for the decomposition and reconstruction steps that were described in Section 3.

Assume that for each n, we select two sets of distinct points x, = {~,k);=~ and y, = {ynk):,: on T with y, such that \kn(y,) of (7.1) is regular. We write in short hand a, = @,(xn) and @, = 9, (y,). In that case p,k(z) = kn(z, x,k) forms a basis for Ln and +,k(z) = kzn(z, ynk) - kn(z, ynk) is a basis for IC, when kn(z, w) is the reproducing kernel for C,. We use the notation of Section 3 and set for f2n E C2n f2n = Pin(P2n = fn+gn with f n = PnVn E Cn and Sn = '-In$, E L. On the other hand, setting 4, = [40, 41, . . . , & I T and 4: = [&+I, $n+ir . . . , 4inIT, then cp, = and +n = \kfiH4:. Hence

Equating coefficients of the corresponding basis functions in fin = f, + g, leads to

This relation allows us to compute p, and q, from p2, and conversely to reconstruct

Pin from Pn and 9,. For example, using the relations pnT = (fin, &) and p,,, = ( f i n , G,,) we find

that 2n

and

In the special case where the v,,k are orthogonal, i.e., when the x,,k are the zeros [n,k of Qn+l(z, r n + ~ ) , then g,,, = (F~,T/(F, , , (E, ,~) and we get for the first of the analysis forrnulas

In general, the are not orthogonal, but if we choose { y , , ) ~ ~ ~ to be the last n zeros {[2n,72+~+r)::0', then by definition &,(y,,) = 6,, , r: s = 0, . . . , n - 1. Thus, the second analysis formula reduces to

For the reconstruction formulas, it follows from pz,,, = (f, + g,, $in,,) that

Again, if the p2n,r are orthogonal, i.e.. wlien the Xnn,k are tlie zeros of Q~ ,+ I ( z , 71n+l), then &n.r = p2n, r /~2n , r (&n, r ) , and the reconstruction forniula simplifies t o

10. Mult i resolut ion analysis

Thus far we have only worked with finite dimensional spaces C,. With our notation V-1 = Lo and V, = C, where n = 2,, we have the nesting

One may wonder what space is ultimately covered wlien s + co. If K is a subspace of the Hilbert space 7-1, then for the {Vk) to form a MRA of K, one should have that

Thus for our rational spaces, we should verify in which spaces the system {Bk)r=o is complete. There are several possibilities.

THEOREM 10.1. ([Ach56, p.244-2461, [BGVHNgBb, Corollary 7.2.41) If CT=,"=,(l - laki) = m then the system {Bk)rzo is complete i n H2(T) (with

respect to the Lebesgue measure) as well as i n H2(T,p) . If JlogP1(t)dQ = -m (t = eZB) and Crzl(l - icrkl) = x then the system

{Bli)?& i s complete in L2(T, p ) .

Jlogp1(t)d6' > -cc is known as Szego's condition while Crzl(l - lalc) = x means that the Blaschke product B(z) = nT=l &((z diverges to zero. Thus the ak: should not approach the boundary T too fast. H2 represents the Hardy space of functions in L2 which are analytic in ID.

In classical multiresolution analysis (MRA) of L2, several other conditions have to be satisfied. Some of these were recently generalized to what is called a second generation LIRA by Swelderis [SwegB]. We shall show however that with a generalization of the definition of the shift operator as it was done for the polynomial case in [FP97] , it is possible to stay close to tlie definition of a classical MRA.

First of all we use the term Fourier transform with respect to the ORF basis. Thus, if f is in cur function space, then we set 3 ( f ) = { f/)r=C=o with ft = (f: 4k).

The scaling property of a classical NRA can be interpreted in t,his general setting by rioting that f E V, ++ suppF(f ) = (0 .1 , . . . , n = 2S), whereas moving from V, to V,+l practically doubles the support of the Fourier transform.

For the shift invariance property, one should show that the basis functions of the resolution and the wavelet spaces are generated by shifts of one father function arid one mother function. To obtain this property, we need an appropriate notion of a shift. To define such a shift, we fix some TO as a reference point on T and define a shift over T as the effect of the shift operator ST via


where D, is a diagonal matrix

In other words --

( s f f k 0 k = 0, 1 , . . .

This definition can be justified as follows. Let us write F ( 0 ) for f (eZe). Then F ( 0 ) is a 2 ~ p e r i o d i c function. In classical MRA for 27r periodic functions, the orthogonal basis is the Fourier basis 4k ( t ) = eZkt , a shift F(B - T ) has the effect that the kth Fourier coefficient of F is multiplied with 4 k ( ~ ) / O k ( 0 ) = e-'liT . Translating this to the functions f defined on 'IT with orthogonal basis 4 k ( z ) = z k , then a shift has the effect that the kth Fourier coefficient is multiplied with 4 k ( < ) / @ k ( l ) = < - l i where < = ezT . So this is just a special case of our definition.

It is now clear that if we define n

(r.n(z) = x d ' k ( ~ O ) $ k ( z ) = k n ( ~ , r ~ ) , k=O

then pnlc ( z ) = k n ( z 7 x n k ) = S z , , k p n ( i ) . Similarly '$,k = S ,,<, $,, where

k=n+l Thus we have one father and one mother function per resolution level and the other basis functions are generalized shifts of these. Consequently, these spaces are shift-invariant, i.e., f E V , + S, f E V,.

The bases used also have a Riesz property, i.e. there are constants A # 0 and B such that

All~nll I l l f I I Bllpnll where the norms are 2-norms and f E V, is given by f = pncpn. It follows from (4.2) and P a r s e d ' s equality that

I f l l L = I l ~ n @ , " ~ n l i ~ = l I ~ n @ , " l l ~

and this implies that

Similarly, it holds that

when f E W, with f = q n @ , This generalized shift operator implies that the functions in the ORK basis

look like (cyclic) shifts of each other. The same holds for the functions in the \VRK basis. We give some examples.

EXAMPLE 10.1. The cyclic shift is illustrated in Figure 3. Which is again our trivial example where cuk = 0 for all Ic and the measure is the normalized Lebesgue measure. If one introduces poles which are close to the unit circle 'IT, then this will have an influence on the form of the shifted ORK and WRI< functions. For example in Figure 4, al = 0.9 while all other cxk = 0. This gives a pole near z = 1.

FIGURE 3. The real and imaginary part of the functions kn(z, w) =

Cz=o(z/w)k (top) and In(.) w) = ~ : E ~ + , ( z / w ) ~ (bottom) for n = 16. All crk = 0. In Figure A, we took w = 1, and in Figure B, we took w = exp( i4~/17) .

A: k16(z, 1) and l16(zll) B: A "shifted" version of A

FIGURE 4. The real and imaginary part of the functions k,(z, w) (top) and ln(z, w) (bottom) for n = 16. Here cul = 0.9 while all other a k = 0. In Figure A, we took w = 1, and in Figure B, we took one of its shifts: w = eZ4"/l7.

A: k16(z, 1) and ll6(zl 1) B: A "shifted" version of A

i.e., near 0 = 0. This pole forces the peak near H = 0 to be more pronounced, but it also implies oscillations of the ORK and WRK functions in the neighborhood of that pole (here near 0 = 0). These oscillations disappear for w = 1 in kn(z, w) and l,(z, w), but they do show when w moves away from the pole. This is almost not visible in Figure 4. However this oscillation effect increases when the pole is given more weight by making it a multiple pole. This implies a visual distortion of the simple shift-like property as in Figure 5. where we took all cuk = 0.9, k > 1. The peak at 0 = 0 is now more pronounced than in Figure 4, and the shifted versions are heavily perturbed and do not look much like being shifted now.


FIGURE 5. The real and imaginary part of the functions k,(z, w) (top) and l,(z, w) (bottom) for n = 16. Here cuk = 0.9 for k > 1. In Figure A, we took w = 1, and in Figure B, we took one of its shifts: w = ei4r"/7.

A: kI6(z, 1) and l16(z, 1) B: A "shifted" version of A

11. S y m m e t r y

We have observed in several of the examples that the real part of the scaling and wavelet functions were symmetric while the imaginary parts showed an antisymmetric property. We can explain this as follows. Suppose that the measure p is real and symmetric in the sense that p(S) = p(3) where S = {e" : 0 5 do 5 d 5 dl I T)

is an arc of the upper half of the unit circle and 3 is the corresponding arc on the lower half circle: = {z : 2 E S ) . Moreover assume that the poles l /Zk are chosen symmetrically, that is the poles used in C, are either real or they appear in complex conjugate pairs. In that case the symmetry that was observed will take place.

THEOREM 11.1. Under the above conditions about symmetry of the measure and of the poles, the kernels kp(z , 1) and therefore also the kernels 12.(z, 1) =

lczs+1 (z ,1) - kzS (z , 1) have a symmetric real part and an antisymmetric imaginary part.

PROOF. Suppose n = 2 S . We first observe that k,(z, w) = CE=o f k (2) fk(w) where {fk)F,o is any basis of orthogonal functions such that C, = span{fk : k = 0. . . . , n). Now consider the measure dp, (t) = dp(t)/ T, ( t ) l 2 where t = e2' and ~ , ( t ) = nz=, (1 - ?tkt) . Define the polynomials pk, k = 0,1, . . . . n by orthogonaliz- ing the functions (1, z, . . . , zn) with respect to p,. Because p, is real and symmet- - ric on T, the coefficients of the pr, will be real. Hence pk(2) = pk(z) . In particular p k ( l ) E R. The reproducing kernel ,&,(z, 1) = c;=~ p k ( z ) p k 0 for the polynomial

space II, with respect to p, will thus satisfy ,&,(z. 1) = ,&,(z, 1). Furthermore, because of the symmetry of the a k , it holds that ~ ~ ( 2 ) = ~ , ( z ) . The theorem now follows by observing that the kernel i,(z, 1) can be transformed into a reproducing kernel for C, with respect to p by setting k,(z, 1) = i ,(t , l ) / [ ~ , ( t ) ~ , ( l ) ] and obviously k,(Z, 1) = k,(z, 1). 0

ADHEhlAR BULTHEEL AND PABLO GONZALEZ-VERA

12. Compu ta t i ons

We have given a recurrence for the reproducing kernels in Section 5. It was however not applicable in its forward form when w E T. There is however a backward algorithm, which is in fact the generalized Schur algorithm of Nevanlinna- Pick. One starts by computing the Riesz-Herglotz transform R of the measure p and from this one computes a Schur function (bounded by 1 in modulus and analytic in D)

where z is the variable and w a parameter such that r ( w , w) = 0. Given the points {ak), one can compute the pk and yk of the recurrence by the following algorithm. Set ro = r, and compute for k = 1 , 2 , . . . the transforms rk = 7 k ( I ' k - l ) where Ik =Ilrc oTLk 0 7 ~ ~ is defined by

The 71;'s and pk's are the parameters that are needed in the recurrence relation of the reproducing kernels. So, given the values of R at the points cuk , it is possible to generate the parameters pk and 3k. However, if some (or all) of the points a k

coincide, then one has to provide the values of derivatives of R in these points and to cover the general case, it is rather difficult to write a tractable algorithm.

A similar argument holds for a Schur-type algorithm to compute the the parameters Xk in the recurrence relations for the ORF as given in Theorem 5.1.

In this text, we were mainly interested in experiments to generate plots for wavelets and see what the influence was of the the introduction of the poles cyk,

we have chosen to use the forward recursion for the ORF for some chosen a k ' s and Xk's and from there compute the reproducing kernels. It is then however not easy t o find what the underlying measure is. If the XI, are all in T, then the existence of a positive measure is guaranteed by the Favard theorem [BGVHN92] . There are several theorems available describing the convergence of the kernels inside or outside the unit disk, but there are however practically no theorems to describe the convergence of the kernels on 'IT.

We do have thc following direct result which is provided by MAt6-Nevai-Totik [MNT91 , Theorem 11.

THEOREM 12.1. Let all a k = 0, so that we are i n the polynomial case. Let 4k be the orthonormal polynomials with respect to a finite positive Bore1 measure d p supported on 'IT. Define k,(t, t ) = l4k(t)I2, t E T. T h e n if (Szegii's condit ion)

J -77

i t holds that n

lim - = pl(t). n+m k,(t, t )

for almost every t E 'IT


We do not know of a proof which generalizes this to the rational case. We hope to provide such a proof in a future paper.

The previous theorem supposes however that logp' is does not vanish on an interval of positive Lebesgue measure. For examples where Szego's condition does not hold, we did observe in our experiments that it does not give the convergence hoped for. Also, convergence is relatively slow, which means that a large n is needed before convergence occurs, so that one may run into numerical difficulties before the computation is complete.

There is however an alternative which is not completely proved either but it turns out to be numerically performing better. We can motivate it as follows.

First recall that if Szego's condition is satisfied then one may define the spectral factor

u ( z ) = e x p { & ~ ~ l o g w ( t ) d ~ } . t - z t =eZe

for z E ID and we normalize it such that a (0) > 0. Moreover it has a radial limit to T such that l ~ ( t ) 1 ~ = pl( t ) a.e. (w.r.t. Lebesgue measure) on T.

For the convergence of the kernels inside the disk we have for exanlple (see [BGVHN98b] or [BGVHNSO]).

THEOREM 12.2. With K,(t) = k,(t, 0)/Jkn0 we ha^

where

Moreover, if Szego's condition holds, and the Blaschke product diverges, then Kn(a) converges locally uniformly to l / a ( z ) in D.

Thus we may hope that plotting IKn(t)l-"or t E T and n sufficiently large will give an idea about the underlying weight.

EXAMPLE 12.1. The previous idea is used in Figure 6. We took all cyk = 0 and chose / A l l = / X z l = 0.4 and all other Xk = 0. We have plotted ks(t, 1) and 18(t. 1) and their modulus. The rightmost pictures show the unit circle with the position of the a k = 0 and below it. we have plotted the approximation for the weight. In the figure B below, we plotted k8(t, 1) and 18(t, 1) multiplied by the weight (pictures at the bottom).

Both of the methods given above give bad convergence when Szego's condition is not satisfied. However. it can be experimentally verified that in those cases we do have convergence of 1 / [2~lS ,1~] with (C, 1) CesBro means

to the weight function w in dp(t) = w(t)do, t = eZe.

EXAMPLE 12.2. We give one last example in Figure 7. Here we took cyl = 0, cyk = 0.9 for all k > 2 All other cyk are zero. For the reflection coefficients we took the same as in the previous example: lXl 1 = lXzl = 0.4 and all other X k = 0. By

ADHEMAR BULTHEEL AND PABLO GONZALEZ-VERA

FIGURE 6. The real and imaginary part and the modulus of the functions ks(z, 1) (top) and ls(z, 1) (bottom) where all a,+ = 0 and IX1/ = IX21 = 0.4 and all other Xk = 0. This corresponds to a weight which is plotted in the lower right corner. In Figure B, we plotted the same functions (real and imaginary parts and modulus) multiplied by the weight.

A: ks(z, 1) and Ls(z, 1)

B: Same functions as in A, multiplied by the weight

comparing with the Figure 6, one can notice the influence of the poles. The weight function is practically unchanged, but the wavelets are much more locallized.

13. Conclusion

We have given a rational generalization of the polynomial wavelets of Fischer and Prestin [FP97] and translated it to the orthogonal rational functions on the


FIGURE 7. The real and imaginary part and the modulus of the functions Ics(z, 1) (top) and ls(z, 1) (bottom) where all cuk = 0.9: except ct.1 = 0, and JXIJ = I X z J = 0.4 and all other XI, = 0. This corresponds to a weight which is plotted in the lower right corner. In Figure B, we plotted the same functions (real and imaginary parts and modulus) multiplied by the weight.

A: k8(2, 1) and ls(z, 1)

B: Same functions as in A, multiplied by the weight

unit circle. The continued fraction-like recurrences and quadrature formulas related to these orthogonal rational functions play a central role.

This results in complex wavelets which can be influenced by the choice of poles near the unit circle.

When one wants to apply this to real 2~-periodic functions. the wavelets being complex is a drawback. Our theory can be generalized to orthogonal rational functions where the points cur, are chosen in a special way like in [BGVHN94,


GVSLN961 where the orthogonal rational functions are obtained by orthogonal- ization of the sequence { B i l , B,:~, . . . , B C ' . ~ . B ~ . . . . , B ~ - ~ , B ~ } .

References

N.I. Achieser, Theory of approximatron. Frederick Ungar Publ. Co., New York. 1956. N. Aronszajn, Theory of reproducing kernels. Trans. Amer. hfath. Soc. 68 (1950). 337-404. A. Bultheel and P. Gonzalez-Vera, Ratzonal wavelet.5 on the real line, Kumer. Funct. Anal. Optim. (1998). Submitted. A. Bultheel, P. Gonzalez-Vera, E. Hendriksen, and 0 . N jk t ad . A Szego theory for rational functzons. Technical Report TW131, Department of Computer Science. K.U. Leuven, May 1990. A. Bultheel, P. Gonzdez-Vera. E. Hendriksen, and 0 . Nj&tad, A Favard theorem for orthogonal rational functzons on the ~ ~ n z t circle, Numer. Algorithms 3 (1992). 81-89. A. Bultheel, P. Gonzalez-Vera, E. Hendriksen, and 0 . Njgstad, Quadrature formulas on the unzt circle based on ratzonal functions, J . Comput. Appl. Math. 50 (1994). 159-170.

[BGVHN98a] A. Bultheel. P. GonzBlez-Vera. E. Hendriksen, and 0. NjLstad, Continued fractions and orthogonal ratzonal functzons. Orthogonal Functions, hloment Theory and Continued Fractions: Theory and Applications (W.B. Jones arid A.S. Ranga. eds.), Marcel Dekker, 1998, To appear.

[BGVHN98b] A. Bultheel. P. Gonzalez-Vera. E . Hendriksen, and 0. Njkstad. Orthogonal rational functions, Cambridge University Press, 1998, (To appear). C. Brezinski, A dzrect proof of the Chrzstoffel-Darhoux identzty and its equzvalence to the recuvence relatzonshzp. J . Comput. Appl. hlath. 32 (1990). 17-25. P .J . Davis. Interpolatzon and approzamation, Blaisdell. New York. 1963, Reprint: Dover, 1975. M.M. Djrbashian. A survey on the theory of orthogonal systems and some open problems, Orthogonal polynomials: Theory and practice (Boston) (P. Nevai, ed.) , Series C: Mathematical and Physical Sciences. vol. 294, NATO-ASI. Kluwer Aca- demic Publishers, 1990, pp. 135-146. W.F. Donoghue Jr . , Monotone matrix functzons and analytic contznuatzon, Springer, Berlin, 1974. B. Fischer and J . Prestin, Wavelets based on orthogonal polynomzals; hlath. Cornp. 66 (1997), 1593-1618. P. Gonzblez-Vera. J .C. Santos-Leon, and 0 . N jk t ad . Some results about numerical quadrature on the unit circle, Adv. Comput. Math. 5 (1996), 297-328. H. Meschkowski, Hilbertsche Raume mit Kernfunktzon, Springer, Berlin. 1962. A. hlat8, P. Nevai. and V. Totik, Szego's extremum problem on the unit czrcle. Ann. of Math. 134 (1991); 433-453. W. Sweldens. The lifting scheme: A constructzon of second generatzon wavelets, SIAhl J . hlath. Anal. 29 (1998). 511-548. G. Szego, Orthogonal polynomzals, 3rd ed. , Amer. hfath. Soc. Colloq. Publ.. vol. 33, Amer. Math. Soc., Providence. Rhode Island, 1967. First edition 1939.

DEPART~~ENT OF C O ~ I P U T E R SCIENCE. K.U.LEUVEN. LEITVEN. BELGIUM E-mail address: Adhemar. Bultheel0cs. kuleuven. ac . be URL:http://www.cs.kuleuven.ac.be/~ade/

DEPARTMENT ANALISIS MATH.. UNIV. LA LAGUNA. TENERIFE. SPAIN E-mad address: pglezQul1 . es

On the explicit evaluations of the Rogers-Ramanujan continued fraction

Heng Huat Chan and Victor Tan

ABSTRACT. In this paper, we discuss recent progress made in the explicit evaluations of the Rogers-Rarnanujan continued fraction via modular equations, singular rnoduli and class invariants. We also introduce empirical processes which will generate neur evaluations of Rogers-Ramanujan continued fraction.

1. In t roduct ion

For Iql < 1. let

denote the famous Rogers-Ramanujan continued fraction, and let S(q) = -R(-q). In his first letter to G.H. Hardy 13, p. 291. S. Rarnanujan asserted that

Hardy was greatly impressed by the beauty of (1.1) and (1.2). 1% quote here Hardy's comments on these identities 17, p. 91:

" T h e s e formulas defeated m e completely. I had never seen anything in t h e least like t h e m before. A single look a t t h e m i s enough t o show tha t they could only be wr i t t en d o w n by a m a t h e m a t i c i a n of the highest class. T h e y m u s t be t rue because, zf t h e y were n o t t rue , n o o n e ulould have had t h e imagina t ion t o inven t them."

The first proofs of (1.1) and (1.2) did not appear unt,il 1929 in a paper by G.N. Watson 1141. Using the quintuple product identity, Watson first est,ablished an important relation (known to Ramanujan) between the Rogers-Ramanujan continued fra~t~ion and infinite products, namely,

He t,llen expressed the infinite products on the right hand side in terms of Weier- strass elliptic functions and deduced (1.1) and (1.2) using the periodicity of the elliptic functions. We remark here, as indicated in a paper by K.G. Ranlanathan

1991 Mathematics Subject Classzficatzon. Primary l lY65

Research funded by Academic Research Fund from National University of Singapore, project number : RP3981645

@ 1W!l !lrrlr ,rica.~ h l . r t l , r ~ r n ; \ t ~ < . , ~ l Sucir ty

128 HENG HUAT CHAN AND VICTOR TAN

[8], that Watson's final arguement can be simplified if we invoke the well-known transformation formula for the Dedekind q-function

where S

q ( r ) = q1'24 ( 1 - q n ) , = e 2 " ~ 7 3

n=1

and Im T > 0. If we substitute q = e-2" in (1.3), then we find that

where the last equality is obtained by substituting T = 5i into (1.4). Solving (1.5), we complete the proof of (1.1). The identity (1.2) can be obtained in a similar way by using the transfo-mation formula for q (F). Hence, with the knowledge of (1.3), the evaluations of the Rogers-Ramanujan continued fraction at q = ec2" and -e-" are reduced to substituting T = 5i into transformation formulas of the Dedekind eta functions ~ ( r ) and TI (+ ) , respectively.

Identities (1.1) and (1.2) are not the only two evaluations Ramanujan gave for the Rogers-Ramanujan continued fraction. In fact, other special values of R ( q ) and S ( q ) are scattered in his Notebooks [lo] and Lost Notebook [ll]. We give a list of these continued fractions, dividing them into 3 classes. Besides the continued fractions listed here, Ramanujan indicates that he could calculate further values of the Rogers-Ramanujan continued fraction but he does not record them [ll, p. 2101.

Class 1A

Values of R(e- ' " f i ) and ~ ( e - " f i ) where n - 1 (mod 4 )

n is not a complete square

Class 1B

n is a complete square

Set 2c,, = 1 + - an + brl fi and R(ep'"") = - c,. a,, - bn

We summarize Ramanujan's continued fractions in the following table

EVALUATIONS OF THE ROGERS-RAMANUJAN CONTINUED FRACTION 129

Class 2

Values of ~ ( e - " f i ) where n - 3 (mod 4)

Class 3A

Values of s ( e p " m ) and ~ ( e - " G ) where 5 + n and n is squarefree n = 1 (mod 4)

Class 3B n - 3 (mod 4)

In the following sections, we will give brief surveys of the continued fractions listed in these classes. We will then indicate certain empirical processes which will enable us to discover new values of the Rogers-Ramanujan continued fraction belonging to these classes.

HENG HUAT CHAN AND VICTOR TAN

2. Class 1

The evaluat,ion (1.6) appeared in Ramanujan's second 1et)ter to Hardy [3. p. 571 and was first proved by 16'atsorl [13]. The key step in \Vatson's proof involves

subst,ituting q = e-'"Ifi into Ramanujan's modular relation

- 1 - 271 + 4'u2 - 31~:' + u 1 , , , , I = ?L

1 + 3 u + 4 u ' + 2 u 3 $111 '

where 11 = R ( q ) and 71 = R ( q " ) . In 1984, Rarrlanat,han [B] gave another proof of t,his evaluation and found the

cornpanion identity

The first cont,inued fraction is identity (1.1) while t>he rest of the continued fractions listed in Table 1 appear on page 311 of Ramanujan's first notebook. The value for R ( e - '"1 was first proved by Ran~mathan using the Rarnanujan- Weber class %nval-iunts

Y

and x

A=I

The proofs of all the continued fractions in Table 1 are now available in a paper b) Berndt and Chan [4]. The identities associated with R(epJT), R(e-"7) and R ( e p l " ) were proved using the following modular relation found in Ramanujan's Yotebook [l, pp. 212-2131:

The proof of Ramanujan's value for R ( C > - " ~ ) is much harder. The first proof given by Bcrnclt and Char1 involves the Rarnanujari cubic continued fraction

q l / : + q2 q2 + q4 q:c + q( i - - - - I + 1 + 1 + 1 + . . . ' M<l.

and the comp~tat~ions are laborious. Not being satisfied with their first proof, Berndt: Chan and Zhang [5] discovered new formulas which express relations between the Rogers-Ramanujan c~rlt~inued fraction and the class invariants G,, and y,, . TJsirig these formulas, they obtained a simpler proof of this identit,y. They also succeeded in proving t2he companion of this ident,ity, namely.

~ ( e - ~ ' ~ ~ ) = JT+7 - c'

EVALUATIONS O F T H E ROGERS-RAMANUJAN CONTINUED FRACTION 131

where n + b

2c = ---h- 1, n = 60'" and h = 2 + h- A. a - h

We now introducr a new empirical process which will enable us to ..guess" all the continued fractions except R(ep") and R(e-lG") in this class. This process is i l lustrat~d with the identity for R(rpG") . Sct

(2.1)

Note that we havc, b y (1.3),

Now, b y cornputiiig and 21, nuniericallj-. we assume that

and

Using these assumptions, we conclude that

It can be shown that (see [ 5 ] ) the value of r:) can be written in t,he form

and Ramanujan's assertion follows. The computation of y ~ , leads to tlle clvaluation of S(rp.").

3. Class 2

The continued fractions listed in Class 2 are recorded on page 36 of the Lost Notebook. In view of the relation [8]

it suffices to evaluate just one of the two continued fractions (1.7) ancl (1.8). The

evaluation of s ( ~ c ~ ~ ) was first proved by Rarnanathan [9] using the valuc


which he found via Kronecker's limit formula. A second proof of this result was given by Chan [6] using one of Ramanujan's modular equations [I, Chapter 25, Entry 621

where

We now introduce the empirical process associated with this class. Let y, be defined by (2.1). We compute the numerical value of ys and assume that

&+I Solving the above quadratic equation yields y~ = -

2 , which leads, by (2.2), to

the value of s(ep"&). The continued fractions we discussed so far were known to Ramanujan. We

now show how this empirical process enables us to discover new simple values for the Rogers-Ramanujan continued fraction which are not found in his notebooks and Lost Notebook.

If we compute yl l and 71, numerically using the product representation (2.1), we may assume that

and

Substituting these values into (2.2), we deduce that

and

(3.5)

These values are new but we do not have proofs for them yet since our computations rely heavily on the assumptions that (3.2) and (3.3) hold.

In the remainder of this section, we will briefly sketch how one can prove (3.4) and (3.5) using modular equations. This method of proof is new and can be applied to give a third proof of (1.7).

To prove (3.4), we first quote a modular equation discovered by L.J. Rogers

[121.

EVALUATIONS O F T H E ROGERS-RAMANUJAN CONTINUED FRACTION 133

THEOREM 3.1. Let u = R ( q ) and v = R ( q H ) . Then

Now, replace q by -q in Theorem 3.1 to conclude that

r where x = S ( q ) and y = S ( q U ) . Next, set q = e-" /" l1 and invoke the relation (3.1) to conclude that

where now z = ~ ( e - " l J i i ) and y = S ( e p a J T i ) . Substituting (3.7) into (3.6) and simplifying using MAPLE V, we conclude that

The continued fraction s ( e p i r m ) is a root of the first polynomial w h ~ h can be used to prove (3.4).

To prove (3.5), one would require a modular equation of degree 19. We have succeeded in showing that if u = R ( q ) and v = R ( q l " ) then

u2" - (21'' + 19v1° + 95v5 + 76)viu1' + 38(4u1° + 45u5 + 4)vJu1'

- 76(v15 + 18u1% 79v" +)v2ui7 + 1 9 ( 1 1 ~ ~ ~ + 168v1(' + 608vi + 4)t1ulb

- 114(11v"' + 121~" 1 2 5 ) ~ % ~ " - 19(v1" 197v1° - 1881u5 - 608)v1u"

+ 76(2v1" 1 0 3 ~ ' ~ ' - 765~" 7 9 ) ~ ~ ~ ' ~ - 114(12v1' - 74v1° - 510v' - 15)v2u1'

+ 19(168v1" 1683~"' - 1881~' - 5)vu1' - 418(33v1" - 79v5 - 33)v5u"'

- 19(5v1'- 1881vH'+ 1683v5 + 168)v3u% 114(15v1" 5 1 0 0 ~ ~ ' + 74v5+ 12)vJuX

- 76(79v1" 765v") + 103~" 2)vLu7 + 19(608v15 - 1881v1' + 197v5 + l ) v u 6

- 114(125u1" - 12121' + l l )u5u ' - 19(4v1' - 608~"' + 168~" 11)v3u4

+ 76(2u1' - 79u1" + 18v' - l ) v J u J - 38(4v1' - 45v" 4)u7uL

+ (7671'' - 95v1' + 19v5 - 1)vu + vLO = 0.

The continued fraction S(eprJiri) can then be determined in exactly the same way

as that of S(eprJ i i ) . Finally, it is obvious that (1.7) can also be proved using the following modular

equation of Ramanujan [2. Chapter 32, p. 17, Entry 31:

( v - u"(1 + uv" = 3 u V ,


where u = R(q) and v = R(y3).

4. Class 3

On page 210 of his Lost Notebook. Rarnarlujan started a table recording t,lic

values of ~(e-"/"') and ~ ( e - " m ) for odd integers between I and 15. Only three continued fractions are actually listed in the table. This table has now been

co~npleted by Herndt, Chan and Zhang [5] using formulas relating s ( c - " ~ ) to GI, and known values for G,,. We now describe an empirical process which will enable us t,o complete Rarnanl~jan's table (except n = 15 which belong to Class 2) without using values of GI,.

Let

(4.1)

The relation between 6,, and S ( r - " m ) is given by [14]

Consequently. it suffices to drterriline 6,) irl order to compute ~ ( e - ~ m ) explic-

itly. The contiriued fraction S(e-'/&) can then he derived from the following reciprocity law of Ranianujan [B]:

1% now illustrate the process for n = 7 and 71 = 9. First, we compute b7 numerically and assume that

Solving the above for d7 and substituting into (4.2). we obtain (1.11). Next, if we assume that

then bg = (4 - 6) (26 + 5&).

From t,his arid (4.2). we deduce that

where

which is equivalent to the result found in [ 5 ] .

EVALUATIONS O F THE ROGERS-RAMANUJAN CONTINUED FRACTION 135

5. Ring Class Fields and the empirical processes

Let K be an irnagii~ary quadratic field and let D1, be the ring of integers of K. Suppose L3 is an order of Dl, with corlductor f and let C'(L3) be the ideal class group of the order D. A ring class field Lo of the order L3 is the unique Abelian extension such that all primes in K ramified in Lo must divide f Dl, and that

We now state some conjectures which we wish to prove in order to provide rigorous proofs of identities such as (3.2) and (3.3).

CONJECTURE 5.1. Let n be (L positive integer not divisible by 5. If n - 1 (mod 4 ) , then r:, and ?;, dejined i n (2. I ) , are real units in the ring class field of Z[5fi]. If 7l E 3 (mod 4) , then 3.: is n red ,unit i n the ring r + m field of Z[:,J-n].

CONJECTURE 5.2. S ~ ~ p p o s e the S ~ ~ ~ L C ~ I L W of the class group of Z[5\/--r)], where n = 1 (mod 4) , as of the type Z , or Z1 + Z2. Then th,e real units mentioned tn Conjectusre 5.1 can be determaned in a j k t e number of steps. If n - 3 (iriocl 3) and the class group of Z [ 5 6 ] is of the type 222 or Zr. then th,e real u n ~ t ?: ccLn be determined i n a fin,ite nun~ber of steps.

CONJECTURE 5.3. Let n be a positive integer not divistble by 5. I f 7, = 1 (mod 4 ) , then A,, and 6,,. defined i n (4.1), are real units i n the ring class field of Z[&%]. If n - 3 (mod 4), then 6,, is a real unit in the ring class field of Z[,,"zG].

CONJEC~TUIIE 5.4. Suppose the structure of the class group of Z [ G ] i s of the type Z4 63 Z2 6 Z2 6 . . . 9 Z2. Then the real units mentioned in Conjecture 5 .3 can be determined i n (I finite number of steps.

If Conjectures 5.1 and 5.2 are true. then we would have rigorous proofs of (3.2) and (3.3) without using modular equations. It also shows wlq we ~hoose 1 1 = 11 and n = 19, as these are probably the only two riumbers satisfying the collditions in these coi1,jectures.

If Corljcctures 5.3 arid 5.4 are true, then we would have proofs of the identities listcd in Class 3. illoreover. we will be able to cornputt, many inore contiiiued

fractions. 1% conclude by showing how we compute ~ ~ ' ( e ~ m ~ ) . 1Ve determine the following identities :

HENG HUAT CHAN AND VICTOR TAN

Using these results, we conclude that

Hence, by (4.2), we can determine ~ 7 ( e - " J 5 7 / 7 explicitly The evaluation of S(q) at this value would be unthinkable if we were to use modular equations or Kro- necker's limit formulas.

References

B. C. Berndt , Ramanujan's notebooks, part 111, Springer-Verlag, New York, 1991. , Ramanujan's notebooks, part V , Springer-Vcrlag, Ncw York, 1998.

B. C. Berndt and R.A. Rankin, Ramanujan: Letters and Commentary, Amer. Math. Soc., Providence, London Math. Soc., London, 1995. B.C. Berndt and H.H. Chan, Some values for the Rogers-Ramanujan continued fraction, Canad. J . Math. 47 (1995), 897-914. B.C. Berndt, H.H. Chan, and L.C. Zhang, Explicit evaluations of the Rogers-Ramanujan contznued fraction, J . Reine Angew. Math. 480 (1996), 141-159. H.H. Chan, O n Ramanujan's cubic contznued fraction, Acta Arith. 73 (1996), 343-355. G.H. Hardx Ramanujan, Chelsea, New York, 1978. K.G. Ramanathan, O n Ramanujan's continued fraction, Acta Arith. 43 (1984), 209 226.

, O n some theorems stated by Ramanujan, Number Theory and Related Topics, Tata Inst. Fund. Res. Stud. Math. 12, Oxford University Press, Bombay (1989), 151-160. S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957. , The Lost Notebook and Other Unpublzshed Papers, Narosa, New Delhi, 1988. L.J. Rogers, O n a type of modular relatzon, Proc. London Math. Soc. 19 (1920), 387 397. G.N. Watson, Theorems stated by Ramanujan ( IX): Two contznued fractzons, J . London Math. Soc. 4 (1929), 231-237.

, Theorems stated by Ramanujan (VII ) : Theorems on contznued fractzons, J . London Math. Soc. 4 (1929), 39-48.

Contcrnporary hlathematics Volume 236. 1999

Absence of Phase Transitions in Modified Two-Component Plasmas: The Analytic Theory of Continued Fractions in

Statistical Mechanics

Dov Chelst

ABSTRACT. In 1961, A. Lenard[9] and S. Prager[ l l ] independently solved the one-dimensional two-component plasma. Their solutions employed an implicit equation involving a continued fraction. In addition, Lenard used the analytic theory of continued fractions to prove the existence of an implicit solution. However, he did not realize that this theory could also be used to prove the analyticity of this solution with the aid of an implicit function theorem.

We have extended Lenard's analysis in [3] to include a class of systems which we call modified two-component plasmas. Any such system can still be described in terms of an implicit continued fraction equation. In this paper. we intend to show that such a solution is analytic. Thermodynami- cally, this implies the non-existence of pressure-dependent phase transitions, i.e. transitions between two phases that occur as pressure P and density p vary while the inverse temperature P remains fixed.

1. Introduction: Statistical Mechanics, Phase Transitions and One-Dimensional Systems

It is not our intent to develop new theorems regarding the convergence or analyticity of continued fractions. Rather, we intend to apply well-known theorems (Theorems 6 and 7) to tackle a problem in statistical mechanics. Thus, this paper will contain no new results for continued fractions, just a new application. However, we believe this application to be quite significant.

What is statistical mechanics? Statistical mechanics studies systems containing a large number of particles. These particles obey a set of microscopic dynamical laws. Starting from microscopic particle interactions, such as pair potentials, one seeks to derive expressions for thermodynamic macroscopic quantities such as the pressure P, as functions of the inverse temperature ,i3 and particle density p. Within

1991 Mathematics Subject Classificatzon. Primary 82B23. 82D10; Secondary 82B05, 82B26, 30B70.

This work was partially supported by AASERT Grant No. AS0281. I would like to thank the organizers of this conference for allowing me to communicate my

ideas to members of the continued fractions community. I would also like to thank Prof. Eugene Speer for discussing the details of this paper's main

argument with me and for scrutinizing my earlier drafts

138 DOV CHELST

this theory, these thermodynamic quantities correspond to either averages or sums over a tremendous number (e.g. lo2') of microscopic variables. The large number of particles is responsible for the statistical nature of the theory.

While developing a thermodynamic portrait of a given system, one naturally asks if it exhibits a phase transition. A collection of particles can behave in different ways: like individual molecules (a gaseous state), like a loosely bound conglomerate (a liquid state), or like a more rigidly bound molecular latt,ice (a solid state). The question yet remains: for a given system, do clear demarcations exist between these different states? Do they represent distinct phases of matter?

A phase transition is characterized by an abrupt change. For example, when we increase the pressure on a gas, while maintaining its temperature, its molecules may condense t o form a liquid. Thus, at a specific pressure, a sudden change in density would occur. Up to this point, the pressure and density would continuously increase upon the gas.

To describe this behavior mathematically, we have a fundamental therrnody- namic relation which describes the system as a function of a "complete set" of thermodynamic variables. One example of such a relation is the average entropy per particle s ( e , p) which depends upon the mean energy per particle e and the particle density p of a system. Another formulation, the chenlical potential (or Gibbs potential) p (P . /3): which gives rise to this article's continued fraction, depends explicitly upon a system's pressure P and inverse temperature 0. All other thermodynamic quantities are calculated as partial derivatives of this fundamental relation. For example, the density p of a system for some P and P is related t o p

by

I 1 I

l ip l l ip2 l ip A phase transition can be defined as a discontinuity in one of p's derivatives.

For example, in the above diagram, a liquid-vapor phase transition of van der Jlraals type can be described by the non-differentiability of a pressure-density curve (isotherm). Below a critical ternperat,ure: there is a certain pressure which corresponds to two distinct densities pl and p2, representing matter in two different phases. For any internlediate value of the density, each phase occupies a proportional fraction of the system's total volume. At either end of this interval along the isotherm, the curve has a discontinuity in its first derivative, i.e. a first-order phase transition.

\iThile one can often prove that a physical system exhibits thermodynamic behavior, one can rarely calculate its fundamental relation exactly based solely upon the knowledge of its microscopic interactions. Thus, to obtain exact results? one needs to examine idealized models. One such idealization involves restricting

CONTINUED FRACTIONS AND MODIFIED TCP'S 139

the analysis to one rather than three dimensions. In one dimension, a number of models have exact solutions.

Rigorous proofs for the non-existence of phase transitions are another rare commodity in statistical mechanics. These theorems, when they exist, focus on showing that some fundamental relation describing a given system is analytic. In one-dimension, the most famous non-existence proof certainly follows this line of reasoning. In 1950, Van Hove[5] showed that one-dimensional systems with only finite-neighbor interactions exhibit no phase transitions.

Plasmas, which consist of charged particles, interact via a long-range Coulomb interaction and are not covered by Van Hove's theorem. Nevertheless, in 1962, Lenard and S.F. Edwards[4] showed, by analyzing the eigenvalues of a differential equation of Mathieu type, that the one-dimensional two-component plasma can exhibit no phase transitions.' This was proven in a way that avoided the continued fraction which featured so prominently in Lenard's original thermodynamic solution of the system.

We can readily generalize the original thermodynamic argument of [9] to include modified two-component p l a ~ m a s . ~ One simply notes that Leriard and Prager's original argument uses a Laplace transform technique reminiscent of the solutions of Tonks and Takahashi for systems with only hard-cores or with only nearest- neighbor potentials respectively.3 Then, one replaces Lenard's Laplace transform of the identity with a more complicated Laplace transform. However, it is not clear how to generalize Lenard's second argument, that proved the non-existence of phase transitions, in so broad a fashion. Luckily, this is not necessary.

2. Modified Two-Component Plasmas

When expressing the thermodynamics of this model, we fix ,!? and the unit charge a. Thus, in order to focus upon the crux of the argument and eliminate distracting notation, we set a = 1 and /3 = 1. We therefore intend to suppress all dependence upon these variables, totally ignoring them in all but two places: at the ends of Sections 4.3 and 4.4.

a a a a - a 1 a 1 1 1 a - 1 - w w w w - .. w w

0 Modified two-component plasmas contain an equal number of positively and

negatively charged particles with unit charges f l and --1 respectively. Each particle resides on the positive half line, and is specified by a pair of coordinates (a,. x,) describing its charge and position. Pairs of particles interact via a one-dimensional Coulomb potential

When considered in three-dimensional space, we view these particles as parallel "charged sheets" that lie perpendicularly to the x-axis and each of which has a L'charge density" a,.

'In fact, he proved it for more general multi-component charged systems satisfying certain conditions.

he entire argument can be found in [3]. 3See [13, pp.48-531. A translation of Takahashi's original article can be found in [ l o , p.251.

140 DOV CHELST

In addition, nearest-neighbor particles, separated by a distance x, interact via a potential $(x). This potential may depend upon the relative charges of the two particles. In this case, we refer to two potentials, $,,,, and $I,,,; as the interactions between neighboring pairs of ions with the same charges and with opposing charges respectively.

The total potential energy H of this system in a given configuration is just the sum of all the pair potentials. Using H, for a fixed pressure P, we employ a statistical ensemble especially suited to one-dimensional calculations, the isobaric- isothermal ensemble4, to determine the fundamental relation p ( P ) exactly. That is all we will say about the basic description of modified two-component plasmas and their statistical mechanical calculations. A full derivation of p(P) can be found in [3]. We will now proceed to the fruits of this analysis.

3. Thermodynamic Results: The Real and the Complex

The chemical potential p of a modified one-dimensional two-component plasma is related to the implicit solution of the equation

for positive P. Specifically, the implicit function z*(P) defined to be the solution of smallest modulus of (2) for fixed P > 0, is related to p by

Through this relation, we will show that p(P) is an analytic function of P. Before we can examine z* we need more information about Q. Q can be

described in two ways:

1 . Power Series: Initially, for fixed P > 0, we define Q as a power series in z about 0,

Each coefficient QN(p) is the result of a statistical calculation involving a modified two-component plasma with 2N particles using the isobaric- isothermal ensemble. As a consequence, each coefficient is a Laplace transform of a nonnegative function5, whose integral converges for all positive pressures P and hence for all complex P with positive real part. Our main concern is that it is positive for all positive P.

2. Continued Fraction: Later in [3], we show that after imposing a few conditions and employing a recursive analysis, a modified two-component plasma's Q can be described by a continued fraction whose partial numerators and denominators are functions of P and z.

The first description shows us that when P > 0, and z > 0 is within the radius

of convergence about 0, Q(P, z) and $ are positive. The second description allows us to fully utilize complex analytic machinery. With it, we can show that Q is a meromorphic function of both P and z. We will see that Q'S continued fraction is

4 ~ e e Percus[8] for a discussion of this ensemble. 5 ~ p , r ( ~ ) corresponds to a portion of the isobaric-isothermal partition function and is thus a

Laplace transform of some portion i N ( L ) > 0 of the canonical partition function.


well-defined when P is restricted to some appropriate open set 0, containing the positive reals R+, while z is allowed to range over all @.

The continued fraction describing Q can be written in the standard form

When $ is charge-dependent, the partial numerators and denominat,ors are quite complicated:

Each q is the reciprocal of a Laplace transform:

- = and O depend upon qopp and vsame through the relations

2 voPP(s) 1

O ( s ) = ( ( s ) ) , and E = ---- (1 - O ( s ) ) . v s a m e V O P P ( S )

When $ does not depend upon the neighboring particles' charges, this description simplifies drastically. +,,,, and $,,, all become equal, while O r 1 and E = 0. Hence,

2 a1 = -2, 9 ( P )

a, = -z,

bo = 0, and

6, = q ( P + n2).

The fact that the description of these continued fractions hinges upon the reciprocal of a Laplace transform, coupled with the desire to have the continued fraction description remain valid for all positive P , motivates us to stipulate the following condition for $.

CONDITION 1. The Laplace transform C ( e P * ) ( P ) determined by any nearest neighbor potential $ describing a modified two-component plasma will converge for all positive P and hence in the complex right half plane Wo = { z : R ( z ) > 0).

This condition is not specific to modified plasmas; it is implicit in the standard thermodynamic description of pure nearest neighbor systems solved by Takahashi[lO, p.25). Thus, Condition 1 is imposed, implicitly or explicitly, on many systems containing nearest-neighbor potentials.

Of course, we must now convince a reader that the reciprocals of Laplace transforms, the q's, are well-defined for appropriate values of P. In other words, we need to show that the Laplace transforms are non-zero. Only then can we show that the continued fractions given by (6) and ( 8 ) converge. We will explore the properties of Laplace transforms which make each 77 well-defined in Section 4.3.2.

142 DOV CHELST

4. The Analyticity of p ( P )

4.1. The Main Result. Now, we will state our main theorem and prove it assuming Lemma 3 that we will prove in Section 4.4.

THEOREM 2. Let a nearest-neighbor pote,ntial $ satisfy Condition 1. When $ is charge-depe,nde,nt, let i t satisfy Condition 8. Let Q(P, z ) be a function that can be described both as a continued fraction (6) and as a power series (4) for fixed P > 0 i n z about 0 with positive coeficients. Then the implicit function p ( P ) deJned via (2) and (3) is analytic for P > 0.

4.2. Proof of Theorem 2. To prove Theorem 2, we must traverse a number of steps.

PROOF. 1. First, we state the lemma:

LEMMA 3. If Q(P,z) satisfies the hypotheses of Theorem 2, then Q is a separately meromorphic function of P and z for all (P, z ) E 0, x C for some appropriate open subset 0, of @ containing R+.

While we leave the proof of Lemma 3 to Section 4.4, we would like to describe its conclusion and give an idea of its proof here. Specifically, we show that if we fix one variable, and examine a neighborhood about any value of the other variable, there is an appropriate kth remainder of the continued fraction, K(a,+k/b,+k), which is analytic in that neighborhood. Thus, the full continued fraction is a rational function of the first k partial numerators and denominators and the remainder term. Since these are all analytic functions of the free variable, Q must be a meromorphic function of the free variable.

This argument utilizes two theorems from the arlalytic theory of continued fractions quoted in Section 4.4 as Theorems 6 and 7. At the same time, it relies heavily upon the asymptotic behavior of Laplace transforms described in Section 4.3. It is not suprising that for a charge-dependent system: we need an extra condition, given in Section 4.4 as Condition 8, to apply these continued fraction theorems. A charge-independent system does not require this additional condition; yet, when $ assumes the proper form, it automatically satisfies Condition 8.

2. An extension of Hartogs' Theorem to meromorphic functions due to W. Rothstein[l2] states that a separately rneromorphic function, e.g. Q, in an open complex domain, e.g. 0, x @, is actually jointly meromorphic. Thus, Q is jointly analytic in a neigborhood of any solution to (2).

3. Moreover, we note that the power series description (4) of Q implies that

3 is non-zero about any solution to (2) provided that both coordinates are positive and z lies in the radius of convergence of Q. Together with the meromorphicity of Q, this assures us that for P > 0, Q'S first pole away from zero occurs on the positive real axis. It also guarantees that , for positive P, a unique positive solution z ( P ) exists within this radius of convergence and that this solution has the smallest modulus of all solutions to (2). Thus, this solution is actually z* ( P ) .

4. We may now employ the complex analytic implicit function theorem[G, 1.B.61 to prove that zf ( P ) is an analytic function of P . Let us briefly review the implicit function theorem.


THEOREM 4 (Implicit Function Theorem). Let f : U 4 C be a jointly analytic function in some open domain U C Cn. In addition, let f (2, z) = 0 for some ( 2 , ~ ) E U , and %(z, z) # 0. Then there exists a neighborhood U 1 about Z and an analytic function g : en-' --t C such that f (2, g(Z)) = 0 and (2, g(Z)) E U for all Z E U1.

This theorem applies t o our situation when we replace f by Q - 1. Since f is meromorphic, it is automatically analytic in a neighborhood of any point that is not a pole of f which includes any point where f = 0. Thus, by the implicit function theorem, z*(P) is analytic in an appropriate neighborhood of P.

5. Since P > 0 was chosen arbitrarily, z* (P ) must be analytic for all positive P . The fact that the solution of smallest modulus z*(P) is always real and positive makes this global result possible. Normally, given two analytic implicit solutions z l (P ) and z2(P) to a single equation, a situation might arise in which zl would have the smaller modulus for some values of P while zz would have the smaller modulus for other values of P. The fact that the solution z*(P) of smallest modulus is always positive implies that for such a switch to take place z l (P ) and z2(P) must coincide for some value of P. By the assumptions of the implicit function theorem, namely that 2 # 0. this cannot occur.

6. The fact that z* (P ) is analytic and positive allows us to conclude that

p ( P ) = ln(z*(P)) is also analytic in P. 0

4.3. Three Features of Laplace Transforms. As Q is defined in terms of Laplace transforms, we should review their relevant features. More specifically, we list three features of Laplace transforms. The first is satisfied for any function with an absolutely convergent Laplace integral. The second is specific to transforms of nonnegative functions (that are not identically zero), which is certainly true in our case as e - i is never negative. The third holds true only for an even more specific situation; this is meant to serve as an example of a Laplace transform's leading-order asymptotics.

4.3.1. Limiting Behavior. If the Laplace transform of a function f , L ( f ) ( s ) =

SoM f (t)e-stdt converges absolutely for some s , it is analytic and defined for all P in some open right half plane W, = {s : %(s) > w}. In addition, limR(,),, L( f ) ( s ) =

0. In our situation, by the assumptions of Condition 1, C(e-i)(s) is defined in Wo and for s E 13, c IHIo, lim,,, Jq(s + n2)l = m.

4.3.2. Transforms of Nonnegative Functions and Their Zeros. For a nonnegative function f that converges in Wo, this limiting behavior is monotonic for real P. In addition, for complex s, Iq(s)l 2 q(%(s)). Moreover, if f is nonnegative and not identically 0, we can be sure that 7 is well-defined in some neighborhood 13 containing R+. After all, C(f)(s) is analytic in MI0 and strictly positive on R+.

Unfortunately, this does not suffice. It is necessary to find a complex neighborhood 0, , containing R+ , in which L ( f ) (s) and C( f ) (s + n2) are nonzero for all n. We will prove that there exists a neighborhood 0, containing R+, on which L ( f ) is nonzero, and which has the property that s E 13, implies s + 6 E 13, for all 6 > 0. In other words:

144 DOV CHELST

L E M M A 5 . Let C ( f ) be a Laplace transform of a nonnegative function f that converges for all P E MIo. I f f is not identically zero, then the zeros of C ( f ) are bounded away from R+ i n any half-plane W, for x > 0.

PROOF. We will show more than the lemma requires. We will show that the zeros of the real part of L ( f ) are also uniformly bounded away from the positive real axis in W , for any x > 0. Given the original function f and a positive real value x , pick some r < i. Now, by simple convergence arguments, one can choose

a constant K so that f f ( t )e-x td t > 0 and

K La (t)e-"dt 5 r 1 f ( t ) e - l td t .

Then, for all sl > x,

Now, simply choose w > 0 so that wK 5 3. For all real values Is21 5 w and 1 0 5 t 5 K , cos sat 2 5 and for complex s = sl + i s2 in the complex strip S, = { s :

W s ) > x, lQ(s)l < w),

The proof is complete when we note that since S, contains no zeros of C ( f ) , the Laplace transform's zeros must all remain at least a distance w > 0 away from R+ in W,. 0

Before we continue, we simply construct an open set using the conclusion of the previous lemma. Given a suitable nonnegative function f and a positive r < 3, define the "zero-less" open set 0, = U,>,, S,. Thus, because L ( f ) has no zeros and is analytic in U,, we are assured that 77 is well-defined and analytic there as well.

Finally, we note that this lemma completes a pair of inequalites. For any S E U,,

4.3.3. Leading Order Asymptotics of C ( f ) - Watson's Lemma. While it is not completely necessary, the inequality ( 9 ) allows us to consider the complex asymptotic behavior of C ( f ) ( s ) as R ( s ) + m. While the limit of C ( f ) is certainly an asymptotic feature, in a charge-dependent modified plasma system, where two Laplace transforms play a role, we need a clearer understanding of C ( f ) ' s asymptotic behavior.

Toward this end, we make a large assumption about f . Every nonnegative function has a set f -'((0, co)) E R+ of points where f is positive. For a function f that has a right-handed limit near the smallest point p in this set's closure, C ( f ) ( s ) ' s asymptotic behavior for large R ( s ) depends very strongly upon this limit. Watson's


Lemma[2, 6.41 covers a number of possible limiting behaviors. While this does not exhaust all possible functions, nor does it include all physically meaningful func- tions6, we will restrict ourselves to those potentials covered by Watson's Lemma. An industrious reader will see that those generalizations of Watson's Lemma that are also found in [2] can be treated similarly.

Watson's Lemma states that if a function f behaves as Ata to leading order as t -+ O f , for some constants A and a > 0 and if L( f ) converges in some IHI,, then

AT(a+l) the Laplace transform C(f)(s) decays as R(s) -+ m, to leading order, like -. (I? is the Gamma function which for positive integers n satisfies I'(n + 1) = n! and using Euler's integral can be written as r ( x ) = Som tx-le-tdt.)This is certainly true for real s and the statement for complex s follows from (9).

In statistical mechanics, we frequently encounter potentials $ that contain hard core exclusions. In terms of f = e-P@, a hard core specifies that f ( t ) = 0 for all t < b for some hard core diameter b. If, in addition, as t --, b+, f behaves as A(t - b)O for some constants A and a 2 0, its Laplace transform L(f ) (s ) decays,

~ l ? ( a + l ) e - ~ ~ ( ' ) to leading order, as R(s)a+l . This follows from Watson's Lemma and an elementary feature of Laplace transforms that can be found in any undergraduate text on the subject. The reciprocal of C(f) in either of these two situations, b = 0 or b # 0,diverges to infinity in an appropriate manner.

We note that while Watson's Lemma also allows for -1 < a < 0, a physical argument disallows these exponential values. After all, since f = e ~ ~ @ , our exponent a depends linearly upon the inverse temperature /3. Thus, while for some values of /3, 0 > a = a1/3 would be greater than -1, there would certainly be values of P for which this would be false and thus the Laplace transform would not even exist. This contradicts Condition I that we imposed upon our nearest-neighbor potential $ in Section 3.

This section on Laplace transforms provides all that we need in this paper. It furnishes analyticity criteria. It gives a complex open set in which all 7's are well-defined, i.e. just the finite intersection of open sets for each separate q. It also provides the asymptotic information regarding our 7's that allows us to apply our continued fractions theorems.

4.4. The Meromorphicity of Q.

PROOF OF LEMMA 3. As stated in the proof of Theorem 2, our aim is to show that some remainder of the continued fraction Q is separately analytic in 0, x @; i.e. analytic in a single variable while the other remains fixed. For this purpose, we employ two theorems found in chapter 4 of the book by Thron and Jones[7, Thms. 4.35 and 4.541. Note that f, refers to the nth convergent of the continued fraction whose value we can obtain by substituting 0 for an+l .

THEOREM 6 (Pringsheim). If / bnl > ianl + 1 for all n , then

i. K (e) converges, and

ii. l f n l 5 1 for all n.

K 6 ~ r y e - p from a potential $(y) = 5

146 DOV CHELST

THEOREM 7. If:

i. a,(x), b,(x) are analytic in some domain D, and ii. SIC1, C2 E @, s.t. for all x E D and n E Z+, f,(x) @ {El, C2, m)

Then K (-1 is analytic in D .

We first note that all the partial numerators and denominators described in (6) and (8) are analytic functions of P and z . In 0,, both qOpp and qsam, are nonzero and hence their ratio is an analytic function. Thus, O and Z are both analytic functions of P. Analyticity in z is even more obvious.

Now, consider O ( P ) and E(P) defined in (7). By the asymptotic behavior of the two q's, according to Watson's Lemma,

/ E ( ~ ) 1 D S ~ v g 2 ~ ' - 2 0 ( r a m t e(buPI'-2bBBIIIII )SO 7

for some constants C and D when s is in 0, and so = R(s). Since ( s ) can never grow asymptotically larger than O(s) , if we wish to prove

that 6, is eventually bigger than a, we must focus on its other term, q,,,(P + n2) which grows like ~ s ~ " " " + ~ e ~ ~ ~ ~ ~ ' " o with so = R(P) + n2. For this to grow more quickly than its competing partial numerator, we impose the following condition.

CONDITION 8. Let $ be a nearest-neighbor interaction within a modified two- component plasma. Moreover: let f (t) = e-Q(t) have the form f (t) = 0 for t < b and f (t) N A(t - b)a as t -t b+ for some nonnegative values b and a. Then, we require that

i. bop, 5 2bsam,, and ii. if bop, = 2bSam,, that a,,, < 2aSam,.

If this condition holds, we are assured that eventually 6, will grow asymptotically larger than a, and the conditions of Pringsheim's lemma (Lemma 6) will be satisfied. After all, whether we fix P or z , the real part of P + n2, that replaces s from the previous discussion, goes off to infinity and our asymptotic analysis holds. The second lemma will automatically follow if we choose any two complex numbers

and (2 with modulus greater than 1. Consequently, Q(P, z ) is meromorphic in 0, x @.

0

REMARK 9. We would like to point out that charge-independent systems do not require the additional condition stated above. If one examines their partial numerators in (8), one notices that they remain fixed as n increases. Conversely, the partial numerators grow asymptotically large regardless of the special form of ec*. Thus, both lemmas provide their conclusions effortlessly.

REMARK 10. In addition, while we have not ruled out temperature-dependent phase transitions, it is possible that a similar analysis will preclude their existence as well. Paying attention to the inverse temperature /3 merely involves replacing P + n2 by P(P + n2) in q's arguments and switching the transformed function from e-* to ecP*. The dependence upon P that arises from multiplying the arguments of q by /3 can be treated in exactly the same manner that we treated P above. Only


the ,8 which figures in the definition of the transformed function itself e c P ~ could prove problematic.

In certain simple instances, e.g., when 1C, - 0 or when $1 is a pure hard core exclusion, temperature-dependent phase transitions are also absent. In those situations, C(eP*) depends on only through its argument P(P -t n2). Thus, in these instances, the line of reasoning described in this paper applies t o ,8 as well.

5. Conclusion

We have accomplished our goal of utilizing the theory of continued fraction to prove a fact regarding modified two-component plasmas. Of course, some questions remain unresolved. Did we use this theory effectively? The theorems employed provided sufficient conditions preventing the existence of phase transitions in these systems. But, how necessary is the additional condition imposed upon plasmas with charge-dependent nearest neighbor interactions. What happens if bopp > 2bSam,?

Continued fractions may yet hold the answer to this question. Continued fractions continue to persist in many models related to the two-component plasma. They appear explicitly in the discussion of one-dimensional ion-dipole systems in [15]. Moreover, they are implicit in every article that calls upon the reader t o calculate an eigenvalue of a Mathieu-type differential (differential-delay) equation with periodic boundary values7 For these reasons, we believe that continued fractions will continue to provide valuable information regarding this family of statistical models.

References

R.J. Baxter. One-dimensional gases with hard-core repulsion. The Phyics of Fluids. 8(4):687- 692, 1965. Carl hl. Bender and Steven A . Orszag. Advanced Mathematical Methods for Sczentists and Engineers. McGraw-Hill, 1978. Dov Chelst. Modified One-Dzmensional Two-Component Plasmas and Generalizations of Schwarr's Lemma. PhD thesis, Rutgers University, 1999. S.F. Edwards and A . Lenard. Exact statistical mechanics o f a one-dimensional system wi th coulomb forces. ii. the method o f functional integration. Journal of Allathematical Physics, 3(4):778-790. 1962. Harry L. Frisch and Joel L . Lebowitz. editors. The Equilzbmum Theory of Classzcal Fluzds. volume 15 o f Prontiers zn Physics, chapter 1-7. W.A. Benjamin, Kew York. N Y . 1964. Robert C. Gunning and Hugo Rossi. Analytic Functions of Several Complex Varzables. Prentice-Hall, Englewood Clif fs , NJ , 3 edition, 1965. Wil l iam B . Jones and W . J . Thron. Contznued Fractzons: Analytzc Theory and Applzcatzons. volume 11 o f Encyclopedia of Mathematics and its Applzcatzons. Addison-Wesley. Reading, M A , 1980. J.L. Lebowitz, editor. Simple Models of Equilabrzum and Nonequilibrzum Phenomena. chapter 1. Elsevier, Amsterdam, 1987. A. Lenard. Exact statistical mechanics o f a one-dimensional system wi th coulomb forces. Journal of Mathematzcal Physzcs, 2(5):682-693. 1961. Daniel Mattis. The Many-Body Problem: A n Encyclopedia of Exactly Solved Models i n One Dzmension. World Scientific. River Edge, NJ. 1993. S. Prager. T h e one-dimensional plasma. Advwnces i n Chemical Physzcs. IV:201+. 1961. Wolfgang Rothstein. Ein neuer Beweis des Hartogsschen Hauptsatzes und seine Ausdehnung au f meromorphe Funktionen. Mathematische Zeztschrift, 53:84-95, 1950. Colin J . Thompson. Classical Equzlzbrium Statzstzcal Mechanzcs. Oxford University Press. London. 1988.

?See [14, 151 and [I] for example.

148 DOV CHELST

[14] Fernando Vericat and Lesser Blum. Statistical mechanics of ion-dipole mixtures: An exactly solvable model in one dimension. Journal of Chemical Physics, 82(3):1492-5, 1985.

[I51 Fernando Vericat and Lesser Blum. Many-body functions of nonprimitive electrolytes in one dimension. Journal of Statistical Physics, 61(5/6):1161-1185, 1990.

DEPARTMENT OF MATHEMATICS - HILL CENTER. RLITGERS. THE STATE UNIVERSITY OF NEW JERSEY. 110 FRELINGHUYSEN RD. PISCATAWAY. N J 08854-8019, USA

E-mad address: chelst0math. rutgers . edu

Some Continued Fractions Related to Elliptic Functions

hlourad E.H. Isnmil and David R. Masson

Dcdzcated to L. .Jerry Lange and Wolfgang Thron on the occaszon of thezr bzrthdays.

ABSTRACT. We study solutions of difference equations associated with the continued fractions of Stieltjes aud Rogers which involve elliptic functions. The denominator polyrlomisls of these .I-fractions arc orthogonal polynomials which arc given explicitly togetllcr with their rneasurcs of orthogonality. These polynon~ials generalize four special cases of thc continuous dual Hahn polynomials.

1. Introduction

In 594 of his book on continued fractions [14], H.S. \.Val1 lists continued fraction expansion formulas for many special functions. His list contains 24 continued fractions. In our generation it was a good exercise for beginners to go through this list and try to prove them all especially after seeing the Gauss continued fraction in 389. Among Iliall's list. five continued fractions are listed undcr group C , titled "Integrals involving the Jacobi elliptic functions." They do stand out as being very different from the rest. Thcy are

where

1991 Mathematzcs Subject Classzficatzon. Primary 30B70, 39A10; Secondary 33E05. Key words and phrases. Elliptic functions, continued fractions, orthogonal polyrion~ials Rcscarch partially supported by NSF grant DMS-9625459. Research partially supported by NSERC grant OGP0005383.

@ 1'399 American hlathrrnatical Society

149

150 M O U R A D E. H. ISMAIL A N D DAVlD R. MASSON

where

(1.6) n,, = (2n + 2)'(1+ k2) , b,, = 2n(271 + 1)'(2n + 2)kL,

where

(1.8) a,, = 2(2n + 1)'(2 - k'), b,, = (2n - 1)(2n)"2n + 1)k4.

IVall [14] attributes the continued fractions in (1.1). (1.3) and (1.4) to Stic1tjr.s [13], and credits Rogers [12] with finding (1.5) and (1.7). A more extensive list of contliiued fraction expansions is in the verj7 rccerit book on the sub jcct by Lorent~en and Waadeland [lo]. Note that the fartor z on the left-hand siclr of (1.5) was left out in Wall [14. (94.20)] and [lo. (2.6.4)].

Obwrvr that the continued fractions (1.3) and (1.4) are equivalent in vicw of the relationship [3. 98.1531. [4. p. 2491

(1.9) CI~(IL. I l k ) = dn(u/k> k).

One can also show that (1.1) and (1.7) are equivalent using Landen's transformation [12]. [15]. Thus, of these five continued fractions. only three arc, independent. n a t ~ i ~ l y (1.1). (1.3) and (1.5).

In Sections 2 arid 3 we will generalize these and derive an additional coritinucd fraction given by

where

(1.11) a,, = (271 + 2)' + (271 + l) 'kL, b,, = (2rtj2(2n + 1)'k'

Equation (1.10) can also be obtainrd from (1.3) in the followirig rnanrier. On the lcft-hmd side of (1.3) 011e can iritegratc bj. parts to get

x

(1.12) J cnju, k) sn(ii. k ) ~ ~ ~ du = k

0

Now substit~lte the right-hand side of (1.3) into the right-hand side of (1.12) and use t 11c idcnt ity

which can be derived from Lemmas I a d I1 in Rogers [12]. This identity should also be cornpared with the identity (3.16) of Section 3 of this work.

SOME CONTINUED FRACTIONS RELATED T O ELLIPTIC FUNCTIONS 151

Alt,llough the result of t,his calculation is indeed (1.10). we prefere the more fundamental derivation of Section 3 , which uses solutions to a differericc equation arid yields both (1.3) and (1.10).

hlotivated by the nlornent problerns considered in [I] and 121. Isinail and Valent [ 5 ] . [6] st,ridied more general nlonient problenls which led to new orthogonal polynomials. The generalization was essentially to replace the lemniscate furictions in [I] and 121 by ellipt,ic int,egrals. This led Ismail and Valent 151. [6] to study orthogonal polyrionlial having generating fimct,ions of t,he form

where g ( t ) is given by

It turns out that thc only choices for rr and 0 arc tr = 0 , -1 /2 and 1) = 0 , 1 / 2 . hence there are essentially four possible sets of orthogonal polynoniials. Tlicse orthogonal polynomials are connected with the denonlinator approxirrlants of the four independent conti~iued fractions (1 .1) , (1.3). (1.5) and (1.10). The case (1.14) with a = / j = 1 1 2 . corresponds to (1.1) while cu = O = 0 correspolids to (1 .5) . We note that cu = -112% 9 = 0 or (r = 0, P = -112 corresponds t o the even denominator approxirnants of (1.3). To see this consider the recurrence relation associated with (1 .3 ) , namely

(1.17) X,,+l ( z ) - z X , , ( 2 ) - n L x , , - 1 ( z ) = 0, n odd.

(1.18) + ( z ) - zX , , ( z ) - ~ 1 ' k ~ ~ , , ( z ) 0 . 11 C V C I ~ .

This yields for n odd

(1.19) X I , + ( 2 ) - [ z L + nL + ( n - 1 ) L k 2 ] ~ , , I ( z ) + ( n - 1)'(r1 - 2 ) ' k 2 X , , $ ( 2 ) = 0.

and for n even

(1.20) X I , + , ( ) - [z' + n 2 k L + ( n - 1 ) ' ] ~ , , 1 ( 2 ) + ( n - ( 7 - 2 ) ' k L X , , ~ ( z ) = 0 .

Each of these call be written as a recurrence relation for rnonic orthogonal poly- rioniials that come from a birth atid death process. The recurrences are of thc form

with birth and death rates A,,, . p,,, , respectively; and s = -2'. The case n = 2171 + 1 is associated with

152 MOURAD E. H. ISMAIL AND DAVID R. MASSON

and corresponds to (2.25) in [6]. since it also has the corrcct initial conditions. On the othcr hand, the case n = 2777 + 2 appcars to be associated with

but it has the wrong initial conditions. To complete the corrrspondence, we note that in (1.15) the case cu = 0. 0 = 0

repeats (] .I) , while a = 0 = -112 repeats (1.5). Finally, the case a = -112. P = 0 or cr = 0. p = -112 1s associated with (1.10) and corresponds to (1.23). which 1s (2.32) in [6].

\Ve will give thesr correspondences in more detail in lattr sections. In Section 2 we derive arid generalize the continued fractiori expansions (1.1),

and (1.5). while Section 3 contains the corresponding gerleralizat~orls of (1.3) arid (1.10). Rogers [12] showed that both (1.1) and (1.5) come from considering the differcrlcc equation for

while (1.3) comes from considering the difference equation for

His principal technique was to usc integrat,iori by part,s to find solutioiis of difference equations.

Having t,hr, integrations in (1.24) and (1.25) e~ t~ending to infinity is. however. misleading. It is 1nlic11 better to consider bot,h end points of integration as zeros of t,he elliptic furlctions in the integrand. The bonus for doing this is that wc are then able to obtain both a dominant and minirnal solution t,o the corresponding t,hree-t,erm recurrence. Thus. in Sect,ion 2. we examine the difference eq~iatioii for the funct ions

2 li

(1.26) C:, (2; k.) = sn" (u . k : ) e z " du. I 0

while in Section 3 we dral with the furictions

2 /I-

(1.27) V,, (z: k ) = / n u k) I ( k ) : ' I d u ;

0

where

Note that z is ROW unrestricted. In Section 4 we calculate the limiting case k -+ 1. This limit consists of four explicit cases associated with special continuous dual Hahn polynomials [8].

Recently Lomont and Brillhart [9] independmtly found the orthogonality relations associated with (1.1) and (1.5) thus overlapping with the results in [ 5 ] , [6].


Alilnc's interesting work [ll] discusses Rogers's method and records related contiii- ucd fractions. This is then applied to study the representation of integers as sunls of squares.

2. A Difference Equation

In this section we consider a difference equations and its ~olutions. This difference equation has two explicit special cases corrt,spoilding to thc coritinucd fractions (1.1) and (1.5).

TVe follow the procedure that Rogers uwd for u, ,(z: k) [12] and ohtdin a generalization of (1.1) dnd (1.5).

Let U,, (2; k) be as in (1.26). Integrating by parts twice, wc have. for 1 1 > 1, n not necessarily an integer.

where in the last line we have used Example 4. p. 516. of [15]. \li, thus have. for n > 1. a solution to the three-tcrrn recurrence rclatioii

(2.1) n ( n + l ) kL Y,,+ 2 (z: k) - (z' + n' (1 + k L ) ) Y,, (z: k) + n ( r ~ - 1) Y,rF2 (2; k) = 0.

Another solution to this recurrence relation is U,,(-z; k) . These two solutions are, however. linearly dependent. To show the linear dependence. we make usc of the properties [15]

(2.2) sn(71 + 2K, k) = e" sn(u. k), sn(-ti, k) = elK sn(u, k).

2 I\

\Ye now rewrite 1 siiri(u, k . ) ~ " " du with a changc of variable (1 = 2K - I ? . and use 0

(2.2) to get

Thus with n = r n + 0. In = 0, k1.1t2, . . . the linear clepcwleilcc is established If wc restrict ourselves to Rc z > 0 and r r an integer. we have anothcr linearlv

dependent solutio~i given h~ u,, (z. k) of (1.21). To see this we split the infinite range of intcgratiori into intervals of lengtli 2h7 arid use (2 2) to obtain

'I\

Hence for n = 0.1 . . . . and Re z > 0 we have

(2.5) u,, (z; k) = u,, ( z ; k) / [l - ( - l ) i ieF 'h2] .

The linear dependence then follows for n even or n odd.


Note that. strictly speaking. U,, (2: k ) , and en"'U,, (z; k) are solutions which are linearly indept:ndmt but in a trivial may. The factor e n T L can be attached t,o any solution of (2.1) to geiierate linearly independent solutions. since, (2.1) is basically a sccond-order dif i r tw;e equat,ion in the square of the shift operator.

Tlic solutions that we have are bounded solutions to the recurrence relat,ion (2.1). The general i d u r e of solutions can be seen from t,he large n as>-rript,otics of (2.1) through the Birkhoff-Trjit,zinsky theory [16]. For n large (2.1) approximates t,he coiistant coeficient tlifftrence equation

(2.6) X.*X,,+~ - (1 + k') X,, + X r r - ~ = 0.

This liab lirlearl>- indepe~ldent solutions

The Birklioff-Trjit z insk~ theory implies that the recursion relation (2.1) will have linearlv independent solutions which. for large n. are asymptotically proportional to the wlutioiis (2.7). With 0 < k' < I wn.c must then have one solution of (2.1) lvith constant asvmptotics and a linearly independent solution wl~icll becomes unbounded as 11 4 30. Since 0 5 m ( u , k ) 5 1 for LL 111 the iri ter~al [O, 2K], we have 11,) (2: k ) bourided as a function of n and corisequentl\ ~t must he a bounded solution

to thc recurreilce relation (2.1) ancl nsymptotic to a coristant multiple of x:," in (2.7).

I11 ortlcr to obtain ari unhoulldccl solutioii to (2.1). we now consider n an intcgcr and

11, '

(2 8) ( z : ) = / [ s ) , n > 0.

- ih '

7r (2.9) 1 ' = ( 1 / 2 , 1 / : 1 1 - k ) O < kL < 1.

2 Tlie integral in (2.8) is in the coniplex u-plane along a contour from u = -7K' to I L = ?I{' . which nrc poles of sn(u. k) , and avoids the pomt rr = 0, wht.re sn(71, k) v;liilsllci. 111 (2.8) t l i ~ point r l = 0 is to the right (left) of the contour whm we use

I;T;y (Cr;,- ) ) . Ti-e again intcgrdtt. b> parts twict, to obtain from (2.8)

valid for rc 2 2. \\.'ith re-liormnlization this becomes the sanw recurrence as (2.1). H e i ~ ( ~ , for iiit(1g~r 71. n > 1. t he rcm~rsion (2.1) has mother linearly indcpcrldent solut ion giwn by

solution\ which is asymptotirally proportioxil to .XLO or ~ 5 ' ) in (2.7). By taking the tliffererlcc of tl-irie two unbounded solutions we obtain a third unboundccl sol~ltion to (2.1) given by


The contour in (2.12) is now a small circle. t,raversed counterclockwiscl enclosing tlic, polr singularity at u = 0. An application of Caucliy's tlleortm sliows that (2.12) is a polynomial in z of degree n - 1 with even powers of z for rl odd and odd powers of z for r1 e w n (see Corollary 2.3). If we refer to (1.2). (2.15) in the work of Isillail a i d Valerlt [6] arid compare t,heir recurrences with our (2.10) we can see that spcciwl cases of the polynoniials in the generating furictions (1.14) and ( 1.15) are given hy

2 . 1 3 ) ( - 2 , 1 / 2 ) = (271 + 1)S , , ( s ; 0.0) = k"+'~:f ,)+, (m: k ) , n > 0. (2.14)

2 (n + 1) k." s,,(.r: 1 / 2 > -1/2) = C , , + l ( r : O . O ) = - - - ~ ~ ~ ) ( J - k s : k ) . ? , > a S J-kr

with the pararnetcrs A and B given bj-

\Ye now summarize these results in terms of thc solutions to a second-ordcr differcrlce equation which is in standard rnonic forin.

THEOREM 2.1. The r e r ~ ~ r w n c r relnflon

(2.16) X , , + I (z; k. cr) - [z' + n,,(k. cr)] X,, ( z ; k , a ) + h,,(k, a ) X , , l ( z : k. a) = 0.

u~rth 2n + C L > 1. 0 < k' < 1. and

(2.17)

o,,(k.. cu) = (277 + a ) ' ( l + k'), b l , ( k . 0 ) = (2n + tr)(2r1 + a - 1)'(2n + rr - 2)kL.

has n mznirnnl solution

Furtherm,ore i f a is c u r i n t q e r then (2.16) h,m domln(mt s o l ~ ~ t i o n s

(2.19) - (+) X!:)(Z; k , a ) = r(2n + n)U,,,+ ,, (z: k ) .

If a = 1 , 2 then (2.16) h>n.s a rnon,ic polynmrninl solution,

PROOF To obtain the recurqiori (2.16) with solutiori (2.18) wc use (2.1) and the solution (1.26) with n replaced 1?\. 271 + a and wt. renormalize. The ~liirlirnal and dominant nature of the solutions comes from thc fact that ULl, +(,(z: k) and - i-t) ULr, +(, ( z : k) have nonzero constaiit large rl as) inptot ic s ( s w the discussion follow~lg (2.7)). Similarly, (2.19) and (2.20) conic from (2.10) drld a renorinalizatio~i of (2.11) a i d (2.12). To sce that the right-hand side of (2.20) is a polynomial of degree. 11 in z' u.1, perform the iiidicated ~ntegration using Ca~chy ' s theorein This is done in Corollary 2.3 below. This tomp1etc.s the proof. 0

Based on the rcsults in Tlieorcm 2.1, we now oht ail1 a continued har t ion expansion which gencdizes ( I . 1) and (1.5).


COROLLARY 2.2. Let a,, ( k , a ) . b , , (k , n ) be given by (2 .17 ) . Then front the m in - imal solution (2.18) to the recurrence relation (2 .16) u ~ e obtain

Un ( z : k ) - b , ( k , a ) b ~ ( k , 0 ) -

1 (2 .21) . . .

I L ( N - ) U p ( : k ) z L + a ~ ( k . 0 ) - z L + a l ( k , 0 ) - z L + a 2 ( k , a ) -

,valid for a > 1 and 0 < k 2 < 1. I n partic,ulrrr, ,we have the special cases

z f sriL(u, k )ep" ' du 1) - -

1 b , ( k , 2 ) bl ( k . 2 ) . . . ( 2 ' 2 2 ) 2 (1 - e x p ( - 2 K ; ) ) z + ( k . 2 ) - z L + a , ( k . 2 ) - z2 + a l ( k , 2 ) -

- - b l ( k 1 ) , , . ( 1 + e x p ( - 2 K z ) ) z L t n o ( k , 1)- z' + nl ( k . 1 ) - z L + a2(k . 1 ) -

These specanl cases m a y be re-expressed as (1 .5) and (1 .1 ) , respectively, after making the additional asswrnption,~ that Re z > 0 .

PKOOF. \Ye apply Pincherle's theorern [7] to the iniriirnal solution ratio

The result is (2.21). In (2.21) we put n = 2 and use the explicit expression U o ( z : k ) = z p l ( l - e x p ( - 2 K z ) ) to obtain (2 .22) . In (2.21) one can take the limit as tr + 1+ to obtain (2.23). Instead we integrate Ul ( z ; k ) by parts twice to obtain the extra recurrenu,

(2.24) Ul ( z ; k ) = - zY2 ( 1 + ep2" ' ) + z-' [- ( 1 + k 2 ) U , ( z : k ) + 2 k ' ~ ; ~ ( z ; k ) ]

Cornbiriing this with the tr = 3 case of (2.21). we get (2 .23) . If Re z > 0 , then we can re-express the left side of (2 .22) and (2.23) by using (2 .5) to obtain (1 .5 ) and (1 .1) . respectively. 0

IVe now give more explicit expressions for the solutions P,, ,, ( z L : k ) , a = 1.2 in (2.20). This will justify our claim that these are polynomials of degree n in z L and the formulas (2.13) and (2 .14) .

COROLLARY 2.3. W e h a w the explicit expresstons


and

PROOF. IYe apply Cauchy's theore~n to the expression ( 2 20) aftcr rearranging the integrand, expanding (u/ sn(u, k))'"+"e-"' in powers of v and finding the coefficient of uL"+ "- ' in t his expansion. Thc fact that sn( u , k ) is an odd function of u causes the odd derivatives of ( u / sn(u, k) )z" t" to vaniqh at i~ = 0. The re5ult i5 (2.25)-(2.28) and the proof of the corollary is coruplete. 0

The continued fraction (2 .21) and the special cases (2 .22) ancl (2.23) are positive definite J-fractions with variable x = 2'. AS such they have a special representation as a Stieltjes transform of a probability measure. Here the measure is discrete so that. for N > 1 ancl O < k L < 1.

with x,,, ( a , k ) real and R,,, ( a , k) > 0. For the special cases n = 1.2. the poles s,,, (a:, k ) and residues R,,, ( N , k ) turn out t o b e explicit. The polynomlal5 PI, ,, ( z ' ; k ) are orthogonal with respect to tlle5e discrete measures.

THEOREM 2.4. Let 0 < k 2 < 1. Then the contim~rcl ,fraction i n (1 .5 ) has (I

Stieltjes tmnsform representation (2.29) ,with rr = 2 and

The contimed fraction i n (1 .1) has a Stieltjes transform representaP~on given b y the righ,t-hand side of (2 .29) with rr = 1 and

PROOF. The singularity structurc of (1 .5) is not directly evident. However if we use (2 .22) we see that the contiriued fraction has pole singularities when ( 1 - e x p ( - 2 K z ) ) / z = 0. This shows that the singularities are poles at 2 = i n , m / K . m = 1.2 . . . . . and yields the expansion


Fro111 t hc k'ouricr serics expansion [I 11. [15]

K - E 27r2 (2.33) snL(n, k) = - - --- cos( r r t~u /K) .

n,=I

wc realize that

Using thest values in (2.32) we obtain (2.30). In the same way we obtain th r expansion

The Fourier serics expansion [15]

is obtained from

0

Using these values in (2.35) we establish (2.31). Tliis completes the proof of Tlieo- rrm 2.4. 0

All of thc above results can be carried over to the case whcn 1 < kL < x if we use the replacement [4, p. 2491

(2.38) sn(u, k) -t k 1 sn(nk, k-I).

Sl'e will not repeat the calculatio~ls for this case. Instcad. we give the malor formula changes. First we cxterid the definitiori of thc U,,'s a?

(2.39) U,, (2; k) = k-" sn" (uk. k ' ) r -"' du /

where

Then th r recurrence relation (2.16) with 2n + (I > 1. but with 1 < kL < x, st ill has a minimal solution

(2.42) X;,')(Z; k, t r ) = r ( 2 n + cu)kL '1~2r ,+ i r ( z ; k.) .


and. if (I. is also an integer, a dorninant solution

(2.43) X?)(Z: k, a ) = r ( 2 n + Q ) ~ ~ ~ ~ ~ ~ ( z : k).

The polynornial solution on the left side of (2.20): with a = 1.2; is now re-expressed as

Furthermore (2.21) is unchanged, but t,he left sides of (2.22) and (2.23) arc replaced

2L/k sn2(uk, k-')e-lLz du.

and

2 L / L

,f sn(uk, k l ) e " ' du

respectively. while (2.30) and (2.31) become

and

(rn - 112)~k"~ 2n2k (rn - 112) pm-1/2 (2.48) s,,(l. k ) =

L , R, , , ( l , k )= - L2 1-p2rll-l '

respectively. Here, p has the same relation to k-' as q had to k. That is

3. The Continued Fractions (1.3) and (1.10)

Since the analysis here is very similar to the analysis in Section 2 , detailed calculations are omitted. We obtain a continued fraction expansion which yields (1.3) and (1.10) as special cases.

Let I/,, (z: k) be as in (1.27). Two integralions by parts show that V,, (2; k) is a solution to the three term recurrence relation

for 0 < kL < 1 and n > 1. Here n is not necessarily an integer. If n is an integer, then additional solutions are given, as in Section 2. by

(3.2) v,,c*) (z: k) = F;-" i dn(u, k)[sn(u, k ) ] l ' l e - U J du,

-111 '


wherc the contour of integration in (3.2) avoids the point 11 = 0. \Vc take the origin to the right (left) of the contour for the + (-) superscript. A linear cornbination of the solutions in (3.2) yields the solution

k-r1 y j L ) ( z : k) = -- dn(u, k) [sn(u, k)] " - ' p - " ' d u ,

2x1 . f which is a polyliolnial of clegrct n in z. These polynomials arc related to thc orthogotla1 polynomials {C,, (x; -1/2.0)) and {S, , (J: -112.0)) in (1.14) and (1.15) by

(3.3)

This is seen by cornparmg the recurrence^ (3.1) with tho5e in [6. (2.24). (2.31)]. rcnornlali~ing. arid applling Cauchy's theorem to thc integral in (3 .3) . The latter stcp i5 don(, below in Corollary 3.3. Notc that Vr,(z: k ) is a boul~ded solutiori to (3.1). whilt~ (3.2) and (3.3) arr uribourldtd 5olutions, as r1 + x.

\Ye now rephrase these result5 in terms of solutions to a threr>-term recurrence relatlon which is in standard ~nonic form.

THEOREM 3.1. The recclrrrnce relaf7on

(3.5) X,,+~(~;k,tw)-[z'+~1,,(k,n)]X,,(z:k,n)+b,,(k.n)X,,-~(z:k,n)=0,

uihere (3.6)

tr,, (k , rr) = (2n + 1 + a)' + (2n + n)'k2. b,, (k , n ) = (2n + n - 1)'(211 + n) 'k2,

w t h 2n + (I > 1 . 0 < I;' < 1 , has a rnzn~rnal solut~on

(3.7) ~ , ( , l ) ( z ; k, tr) = r ( 2 n + 11 + I ) ~ ' ~ ~ V ~ ~ , + ~ , (2: k)

If cr 1 5 an znttycr then ( 3 . 5 ) has dorn~nant sollitions

(3.8) X;')(z: k. n ) = r ( 2 n + a + l)kL"~L(:l,, (2; k) .

If a = 0 , 1 then (3.5) has a morizc polynom~al solution

which 6s of drqrer n in z2

PROOF. 111 t,he recurrcrice relation (3.1) with its solutions V,, (z: k ) . V,!")(Z; k) ,

L',ja(a; k) ill (1.2'7). (3.3) and (3.4) we replace 11 bj- 2n + tr and renormalize to obtain (3.5) wit,li solutions (3.7). (3.8) and (3.9). Tht, bouridedlicss of V2,,+,,(z; k)

shows that x(,' ' ( a : k) is miniirial as compared wit,ll X,(,*) (z; k ) , which is dominant

because v,(,'/,, ( z ; k) is ~mbou~ided. The pol>-nomial nature of Q,,,, (z" k) is rnatle cxplicit in Corollary 3.3 below. This complct,es the proof. 0

COROLLARY 3.2. Define a,, (k. cr), b,, (k , n) thrm~gh (3.6). Then frorn the min- zrnal sokution (3.7) to the rec~/.rrrnce relntzon (3.5) we obtuirl th,e contir~ued fraction expun.sr;on

V , ( 2 ; k) 1 bl ( L 0) b~ (k. n ) (3.10) - - - . . .

cu(a - 1)V,, L ( ~ ; k ) z L + c ~ , , ( k . . n ) z2 + n l (k . (1)- zL + a ~ ( k , 0 2 ) -


which holds for cu > 1 and 0 < k2 < 1. For a = 0: 1 there are th,e additional special case.5

2 11.

J cn(u, k)eP"' du - -

1 (3.11) I'

bl (k. 0) b2(k. 0) . . . z(1 + exp(-2Kz)) z' + ao(k. 0)- z' + a l (k . 0)- z' + a2(k. 0)-

If Re z > 0 then (3.11) and (3.12) ccm be reexpressed as (1.3) cmtl (1.10), respec- t ively.

PROOF. To obtain (3.10). apply Pincherlr's t l-~eorem for the minimal solution ratio

x/ , ' ' (z : k . n ) / [b,,(k.n)x!/(z: k .o ) ] .

To derive (3.11) wc integrate In?- parts twice to calculate the extra recurrence relation

(3.13) k) = z-' (1 + pp- ' " ' ) - z-' [&,(z: k) - 2 k L v ~ ( z ; k)] . and combine this this with the case a = 2 of (3.10). Similarly we have the additional recurrence relation

which, when cornbined with the case cu = 3 of (3.10) yields (3.12) To obtain (1.3) and (1.10) we use the relation

2 I\

f cn(u, k)[sn(n, k)]" e"'C111

m(11. k)[sn(u. k)]" r>-"'du = i,

1 + (-1)" exp(-2Kz) '

I I

The left-hand side of (3.12) can then he rrplaccd by cn ( r~ . I;) sn(rr, k ) e "'drr and 1 ( 1

(1.10) results. To rstablish (1.3) urc> use (3.15) with n = 1 on the left-hand side of (3.11) and we re-exprcss the rlght-lland side of (3.11) using the identity [12. Lenliria

I1 1141

This completes the proof. 0

Just as in thc case treated in Scction 2, we now give explicit exprcwions for the orthogonal polynomials wliich are the drnoniinator approximants of thc continuc~l fractions in (3.11) and (3.12).


COROLLARY 3.3. W e have

(3.17)

whew

(3.18)

and

(3.19)

with

(3.20)

(2n)! d2~~-21 7 1

E7 ,I = ( 2 ~ ) ! ( 2 n - 2 j ) ! d u ~ ~ ' - ~ ~ - [ ( ) 2 7 ' 1 sn(u,k.) d n ( u k)] 1 , 71 =O

PROOF. Apply Cauch?-'s theorern to (3.9). 0

Our next theorem describes the Stieltjes transform associated with the continued fractions in (3.11) and (3.12).

THEOREM 3.4. The continued fractions i n (3.11) and (3.12) have the Stieltjes trnn.sforrn representations

PROOF. We expand the lcft-hand sides of (3.21) and (3.22) in terms of poles and residues using the zeros of 1 ZIZ exp(-2Kz) and the values

i 2 d mq"' (3.24) cn(u, k) sn(u, k) \in(n~.iru/K)du = ----

kL h 1 + qL11' '

I I


for n, = 1 , 2 . . . . . Thesc values are equivalent to thr, Fourier series expansions [15]

since (3.26) the left-hand side of (3.26) is the derivative of d n ( l ~ , k ) . The polynomials Q,, o(zL; k ) and Q r i 1 (zL: k ) are orthogo~lal with respect to the

discrete measures in thc right-hand side of (3.21) arid (3 .22) . rrspectively. In a manner similar to the case in Section 2, we can extend these results to

the parameter range 1 < k L < m. This is done through thc replacerncnt (2.38) toget her with the replacement5

We leave the details as an exercise for the reader. 0

4. Limiting Cases

The results of $2 and Q3 can he extended to the limiting case k = 1. Except for Theorems 2.4 and 3.4, this extension easill- follows from the fact that

(4 .1) lim sn(u, k ) = tan11 u , lirn cn(u, k ) = lim dn(u, k ) = sech 1 1 . A-1- k - 1 A - - 1 -

The limiting case k + 1 of the continued fraction expansions (1 .1 ) . (1 .3) . (1 .5) and (1.10) then give continued fraction expansions for the Laplace transfornis of t an11 u . sec 11% tanhL u , and tanh u sec u, respectively. Howcver. the Sticlt jes transform rcp- resentation of these continued fractions now have absolutel) contmuous measures. In this section we show how to perform this limiting process.

Recall that A' and h" are specla1 hypcrgeoinetric functions of argunlcnts X" and 1 - k 2 , respectively. see (1.28) and (2 .9) The\- are rclatcd to q through thc important formula

[ 1 5 , p. 4861. %'it11 . r , , , ( tr , k ) , R,,, ( ( 1 . I ; ) . a = 1,2. as in (2.30). and (2.31) and f ( x ) = l / ( z 2 + .r) we find

(4 .3)

In view of (4.2) and (2.31) wc find

Forrnulas (1.28) and (2 .9) show that as k + 1 the mcsh .rr,,, I p ( 1 . k ) - .r.,,, . / . ( I . X ) tends t,o zero. Siricc K' + 7r/2. it is a st,andard argument to now show that the riglit,-hand side of (4.3)

MOURAD E. H. ISMAIL AND DAVID R. MASSON

Similarly

(4.6)

a s k + 1. IVe thus arrive at the following k + 1 limits for (1.1), (1.5). and their Stieltjes

trallsforin representations

The discrc,tc measures in Theorem 3.4 are dealt with in the same mariner so that thc k -. 1 limits of (1.3) arid (1.10) can be written as

Note that the Stieltjes transform rcpresentatioiis of the continued fractions in (4.7)- (4.10) are valid for the full convergence region Rez # 0 while the Laplace transform representations are valid only for Rez > 0.

These four cascs correspond to special cases of continuous dual Hahn polynomials. The coinparison with Section 1.3 of Koekoek and Swarttouw j8, p. 271 shows that (4.7)-(4.10) are associated with continuous dual Hahn parameters


Using (4.11) and [S. (1.3. I ) ] we can now statc the <FL forinulas for the d m o ~ n - inator polynomial approxiinants of the continued fractions in (4.7)-(4.10). They are

(4.14) Q,, ~ ( 2 ' : 1) = (1/2),, n! 4" jFL

and

(4.15) Qr, 1 (zL: 1) = (3/2),, n! 4" (Fr

respectively.

References

[I] C. Berg and G. Valent. The Ncvnnlzn,na parnmeterzzatzon for some mdeterrnznate Stzelt~rs m,ornent prob1em.s assoczated wzth bzrth and death processes, hIethods and Applications of Analysis 1 (1994), 169 209.

[2j Y. Chen ant1 hI. E. H. Isrnail. So~rie mdetermznate n~omen t problem,^ and Freud-lzke umghls, Constructive Approxi~rlation 14 (1998), to appcar.

[3] I. S. Gradshteyn and I. hI. Ryzhik, "Tables of Integrals. Series. and Products," Acatlernic Press. Orlando, 1980.

[4] H. Hancock, Lectures on the Theory of Elliptic Functior~s, \'olume 1: Analysis, Wiley. New York, 1910.

[Fi] h1. E. H. Ismail and G. Valent, On n f am~ly yf orthogonml po lynom~ul~ related to cllzpt~c fun,ctzons. Illinois J. Math., t o appear.

[6] hl. E. H. Issnail and G. Valent. Some orthogon,al polynminmls related to cllzptzc func.t7ons. to appcar.

[7] W. B. .Jones and W. J. Thron. "Continued Fractions: Al~alytic Tl~cory anti Applications," Addiso~~-Wesley, Rearling. h l h , 1980.

[XI R. Kockoek and R. Swarttouw, The Askcy-scheme of hypcrgcornctric orthogonal polynomials ant1 its 9-ar~alogues, Reports of the Faculty of Technical hlathcrnatics ;md Infor~nation No. 94-05. Delft University of Technology, Delft. 1994.

[9] .J. S. Lo~norlt autl J. Brillhart. Certain Elliptic Integrals anti Their Orthogonal Polynomials. to appear.

[lo] L. Lorentxcr~ and H. Waadclar~tl, Tent inucd Fractior~s With Applications." North Hollaml, Arnstcrtlam, 1992.

[ I l l S . C. hIilnc. S u m s of s p a r e s . .Iacobz ell~ptzc functzons an,d continued fmctzons. and Schur functzons, t o appear.

[I21 L. J. Rogers. On the representatzon of certain asymptot~c serzes as convergent contmued fractzons. Proc. London Math. Soc. (21, Vol. 4 (1907). 72 89.

[13] T. J . Stieltjcs. Sur la r6duction en fraction continue d'une skric prkc6tlent suivant les pouis- sarlccs descerdants d'une variable, Ann. Fac. Sci. Toulousc. Vol. 3 (1889), H. 1 17; Oeuvres. \b l . 2 , 184 200.

[14] H. S. Wall, Analytic Theory of Continued Fractions. Van Nostrand, prince tor^. 1948. [15] E. T . Whittaker and G. N. \t;atson. "A Course of hIoderr~ Analysis." Fourth Edition. Carrl-

bridge University Press, 1962. [lfi] .T. Wimp. "Cornputation Wit21 Recurrerice Rolatior~s." Pitman. Boston. 1!1X4.


DLPART~IENT OF ~ I A T H E ~ I ~ Z T I ( ' S ~ N I \ ERSITY OF SOLJ I H FLORIDA TAMPA E'L 33620-5700 E - m a d address ~smailQmath. usf . edu ~ E I ' A R T ~ I E N T 01: .\IATH~;RIA.I'ICS, UNI\.ERSITY OF TORONTO, TORONTO. ONT..IRIO, CANADA

115s 3G3 E - m a d address: massonQmath. toronto. edu

Contemporary Math~rnat ics Volurrlr 236. lX49

Asymptotics of Stieltjes Continued Fraction Coefficients and

Applications to Whittaker Functions

William B. Jones and Guoxiang Shen

Derlzcated to Professor L. .I. Lange on the occaszon of hzs 70th bzrthday.

ABSTK.ACT. This work extends recent results obtained by W. B. Jones and W. Van Asschc for a class of S-fractions that inclndcs the Binet function and hence the Garnrna function. Functions subsurned by the present paper include Whittaker function IV, ,,(z) and, in particular, Bessel functions

The asymptotic behavior of the coefficients in the S-fraction expansions of these functions is derived from the Freud conjecturc proved in 1988 by D.S. Lubinsky, H.N. hlhaskar and E. G. Saff. Knou~ing the asymptotic behavior of the coefficients enables us to determine whether or not t,hr S-fractions are convergent arid, when convergent, t o give the asymptotic speed of convergence.

1. Introduction

Many special functions of importance in mathrulatical and phi-sical sciences can be expressed in terms of Stieltjes transforms

where y is a positive measure on the half-line (0, m) with moments x

(1.2) c i = 1 t"dv(l). 1 = 0.1.2 . . . . .

For corwenience we call such functions S-functions and note that all S-fu~lctions are analytic in the cut plane

An S-function G(z) has an asymptotic power series expansion (as z 4 m. z E A

S,, := {z E @ : arg zl 5 a} . where 0 5 a < T ) given by the formal Laurent series

1991 Mathematics Subject Classificatzon. Primary 30B70, 30E05, 30E10, 40A15, 41A20

Research supported in part by the U S National Science Foundation under Grant No DhIS 9701028

168 WILLIAM B. JONES AND GUOXIANG SHEN

Some examples of filnct,ions that can be expressed in terms of S-funct,ions include the Binet function J ( z ) (where T ( z ) = f i z z - + ep'e . ' i t ) ) , polygarnrna functions ~ ( " ) ( z ) and \Vhit,taker functions 1Ir,,,,(z), which subsume other special functions such as exponential integrals E,, ( 2 ) . the error function erf ( z ) . modified Bessel functions K,, ( z ) , and incornplete gamrna functions r ( n , z ) (see, e.g., [ I ] . [5] , [7]. [8] , [ lo] , [12]. [16] , [17]) .

It was shown by Stieltjes in [13] that, for every S-fiinct,ion G ( z ) , t,here corresponds a unique Stieltjes continued fraction (S-fraction)

A bi 62 b:j b~ G ( z ) := -

z +i+t+i+ . . . , b,, > 0 for rr E N

such that. for each n E W := { 1 . 2 , 3 . . . . }, the nth approxirnant y, ,(z) of (1 .5) is a rational function, analytic at z = x, and g,, ( z ) has a Laurent series of the form

convergent in a neighborhood of z = x. illoreover, if an S-fraction (1 .5) is conver- gtnt for each z E S,. then it represents the S-function G ( z ) on S,. Stieltjes showed that an S-fraction (1.5) is convergent if arid only if

It is readily seen that (1 .7) holds if {b,l)z=l is boundtd. Other conditions sufficient to ensure that (1 .7) holds are:

= x (Thron's criterion [14]) rr=l

and

(1.9) = x (Carleman's criterion [ s ] ) . I, = I

Thus it can be seen that convergence of an S-fraction (1 .5) dcpcnds upon asymptotic pr~pert~ies of the cocfficicnts b,, as n + x.

The primary purpose of the present paper is to dctcrrninc asyinpt,otic properties of the br, in (1 .5) in those cases for which there is no knowri closed-form expression for b,, in terms of n,. Our consideration is re~t~rictecl to ~ituat~ions in which the measure y in (1.1) is absolutely continuous:

where

(1.10b) c ( t ) := ~ ' ( t ) , for

hence we can write

for z E ST:

ASYMPTOTICS OF STIELTJES CONTINUED FRACTION COEFFICIENTS 169

IVe use properties of the nonnegative weight function 7 ~ ( t ) in (1.10) to deduce asymptotic properties of the sequence { b , , } ~ = , . Ehllowing the approach recently employed bv Jones and Van Assche in [9], we make use of the Freud corljecture/tlleore~n proved in 1988 by Lubinsk>, hIllaskdr and Saff [ l l ] . For later reference that theorem is stated in Lemma 2.3 and is referred to hereafter as the FLMS-theorrrn. The main result given by Jones and Van Assche [9] involves the Bin& function J ( z ) defined by

. for z E S,,

By using the FLilIS-theorem and properties of the weight function u,(t), Jones and Van Assche proved that the coefficients bd of the S-fraction representation of G J ( z ) satisfy the asymptotic condition

b1 1 i n = - (corijectured by Cizek and Vrscay in 1982 [3])

n - * x n 2 16

The present paper extends the work of [9] by deriving the as? rnptotic behavior of S-fraction coefficients b,, for a large class of S-functions which subsumes (1.11) as a special case. Our main result (Theorem 2.1) is stated and proved in Section 2. Section 3 is used to describe applications of Theorem 2.1 to functions a,, i (z) . closely related to Whit taker functions ti',, ,, (z) and modified Bessel funct ions h;, (2).

Before proceeding to Section 2, we sulnrnari~e (Theorem 1.1) some kilown rc- slilts on convergent S-fractions. which rclate asvnlptotic propertics of thr, S-fraction coefficients b,, (as n 4 x) wlth asymptotic properties of the nth approxiinnnt g,, (z) (as z -+ x and as n 4 m ) . These resiilts can bo found in [8. p. 1951 with a srnall nlodification for Theorem 1.1 (C) .

THEOREM 1.1. Let G(z) be an S- fraction (1.5) ,which conve~yes locally ,!mi- forrnly on S, to an S-function (1.10) (or more generally (1.1)). Let b,, denote the n t h coeflcaent and g,, (z) the n t h approximant of G(z). Let z E S,. Then:

(A ) Suppose lirn,,,, b,, = 0 and let S he a subset of @. If th,ere exist constants p > 0 an.d p > 0 such, that

lb,, 1 5 inf lzl nlin 2 t . S

for (dl n t N. [ F ' $1 then thrrr crrst constants A > O and B > 0 (zndependrnt o f n and z) such that

(13) Suppose lirn,,,, b,, = b and b / z E C\( -m. - f ] for all z E S where S C @. Then for

there exzsfs a constant h7(q). d e p e n d e n t of n and z, such that

(1.16) s i l p ~ G ( z ) - y , , ( z ) / < K ( q ) c ] " , for n € N . 2 E S


(C) Let 0 E R satzsfy 0 < / O < T and let b,, = O ( n 7 ) as n 4 cxj for s o m e constant 7 uizth 0 < 7 < 2 . T h e n there e . r~s t constants A > 0 , B > 0 and C > 1 (zndependent of n, and z ( but dependent upon 0 ) such that for all n E N, (21 > 0 and I arg 21 = 0 ,

, d 7 . = 2

, , . zf 0 < 7 < 2 , where 6 ( 2 ) := ( 1 - i) L/i;T.

2. Main Theorem

Before stating the main result of this paper it is convenient to introclucc a family of fiinctio~is, dcnotecl by Q , and defined as follows: A function Q : R - R is a mernber of Q if Q ( z ) satisfies the following conditions:

( 2 . l a ) Q ( - s ) = Q ( x ) for all r E R (i.e., Q is even):

( 2 . l h ) Q 1 ( r ) exists for .x > 0:

there exist const,ants A I > 0 and E > 0 such that

( 2 . 1 ~ ) I x Q 1 ( r ) 1 5 hI for 0 < r < E :

there (,xist constants X > 0 and B > 0 such that. for all 3. > N .

and

z Q 1 I ( x ) linl ---- exists. -+.; Q1( . r )

THEOREM 2.1. Let positive constants a , 6 a n d c be given. Let Q ( x ) be a nzernber of the family Q that .satisjies the additional conrtitions:

( 2 . 2 ) Q ) i s con t i r~uous for all s > O

and ei ther (Cast: a )

( 2 . 3 ~ ) Q / ( J ) = CS"- + ~ ( ~ r l - l - h ) as x - - t+m

o r (Cnse b)

(2 .3b) rv > 1 a,nd Q 1 ( s ) = cs" - ' + o ( l ) , as x 4 +x.

Let u:(.r) and ? j ( t ) he defined by

( 2 . 4 ~ ~ ) 14) := , for - 3(j < .r < x (1 n rl

T h e n : ( A ) T h e mornenis X

( 2 5) ( J ) nlrd c l1 z ~ x t l l l > ( t ) d t , n = 0 . 1 , 2 , . . . ,


(B) Let G ( z ) denote the S-function defined by the S- tmnsform (1.10): let L(z) A

denote the formal Laurent series (1.4); and let G(z) denote the corresponding S- fraction (1.5) with coeficients b,, . Then

(C) Let N satisfy N > 1. Then the S-fraction G(z) converges locally uniformly on S, to G(z) and the asymptotic behavior of the n t h approziw~ants g,(z) of G(z) is described by (1.7) of Theorem 1.1, since b,, = 0(n2 / " ) as n --t X . and y:= 2/a satisfies 0 < y 5 2.

Our proof of Theorem 2.1 makes use of two important lemmas whose proofs can be found in references cited. They are stated below to make the present paper self-contained. The first of these results (Lemma 2.2) has been proven in [9], but it was not stated explicitly there. See also [2. Theorem 9.1 of Chap. 11 for a partial proof.

L E ~ I M A 2.2. Let ~ ( t ) be a non-negative weight function on (0, no) whose moments c,, i n (2.5) exist. Let e ( z ) , L(z) and G ( z ) d e n o t ~ the corresponding S- fraction (1.5), formal Laurent series (1.4) and Stieltjes transform (1.10). respectively. Let w(x) denote the even weight functzon defined by

(2.8) w(x) = JxJu(x2), for - 00 < x < 30 Let { p , , ( ~ ) ) : = ~ , denote the orthonormal polynomzal sequence wzth respect to ul(x) and let {a,,):=, denote the coeficzent sequence for the three-term recurrence rela- tzons

Then the coeficients b, i n the S-fraction (1.5) satisfy the relations

(2.10) b,, = n i p , , n = 2 ,3 ,4 , . . . .

The second known result used in our proof of Theorem 2.1 is the FLhlS-theorem (Lemma 2.3) which verifies Freud's con,jecture; it was proven by Lubinsk~-, Alhaskar and Saff in [ll] (see also (151).

THEOREM 2.3 (FLAlS-theorem). Let w(x) be an e v m wezght functzon on (-X, m) of the form

(2.11) w(T) = e - Q ( l ) for - c x , < x < m

where Q(a) is a member of the family Q (see (2.1)). let { p , , ( ~ ) } ? = ~ denote the orthonormal polynomial sequence with respect to w(x) and let { n , , ) ~ = = , denote the coeficien,ts zn the recurrence relations (2.9). Let q,, denote the positzoe root of the equation

a,, 1 lim - = - , I +" q,, 2 '


PROOF. Proof of Theorem 2.1 (A) C a s e a . Suppose that (2.3a) holds. It follows that there exist real constants B > 1, A l l > 0, such that , for all s > B > 1.

(2.14) Qf( . r ) - cxnp1 AI] . F - ~ .

Clearly there exists a constant B' > B > 1 such that

, a - 1 - h < :,y-1 1 .l for . c 2 B 1 .

- 2 It follows that

c < , - I 3c - 5 Q1(x) 5 ,.I-"-'. for .r > B' 2

and lierice therc cxists B" > B' > B > 1. such that. for x > Bff .

Therefore, for cach integer n > 0, there exists a number B, > Btf such that

(2.16) e Q ( ' ) > - x"+', for x L:> B,,;

hence

F'rom (2.la) and (2.17) one can con( lude that the moments p,, in (2.5) (with respect to w(x) ) exist. From this aiid (2.4b) it is readily shown that the lrlomerlts c,, in (2.5) with respect to ~ ( t ) exist and 5atisfy equations (2.6). By a si~nilar argument one can show that (A) holds if (2.3h) is satisfied. This concludes our proof of (A).

(B): Canr 0 . Suppose (2.3a) holds. A similar argument holds if we assume (2.3b) instead of ( 2 . h ) . \Ye begin In\- showing that

whcrc q,, denotes the positive root of (2.12). Our proof is by contradiction. Hence we nssumc tlidt (2.18) is false. It follows that tlirre cxlsts a corivergent subseyuen~e

{qIlA )EO of {(I,, ):=:=(, and we write

(2.19) q := lirn q,,, < w. h - x

Since Q1(x) is continuous for x > 0 and ( 2 . 1 ~ ) holds, there cxists a constant AI > 0 such that

d s 7l It is casilv seen that - . B\ applying Lehesgue's dominated comer-

gencc thcorcrn [6] to (2.19) we obtam

whicl~ contradicts the fact that limb ,, r l h = m. U'e conclude that (2.18) holds. From (2.3a) we obtain, for 0 < a. < 1.


It follows froin this and (2.18) that

Applying this and the Lebesgue dominated convergence tlleorem [6] to

yields

The desired result (2.7) follows from (2.10). (2.13) and (2.23). This completes the proof of (B).

(C) is an immediate consequence of (2.7). Thron's criterion (1.8) arid Theorem 1.1 (c').

3. Application to Whittaker Functions

S'i'hittaker's differential equation is given by

where r; and p are real parameters. If 2p + 1 is riot an integer. a system of linearly independent solutioris of (3.1) is given by

(3.2a) Al, , ,, (z ) arid Mh , , (2).

is called the Whzttaker functzon of the first k m d . Here 1 F1 is the confluent liyper- geometric function

where a and b are complex parameters with b 6 [O, - 1. -2, -3, . . . ] and where ( a ) ( ) := 1 and ( a ) A := a ( a + 1) . . . ( (L + k - I ) , for k = 1,2.3. . . . . The branch of ~ ' ' ~ 4 in (3.2b) is chosen so that the branch cut is on the negative real axis arid s t l + f > 0 for -(-. > 0. (See. e.g.. [I] , [5], [li']. for more on Slrhittaker functions.) Wh?ttcrker',s functton of the second kznd It', , , ( z ) is defined by

whcre IT7+, ()(z) := l i~n , ,_ ,~ LITh i, (z). In this section we apply Theorem 2.1 to the functions \Ir, ,,(z). Following Hcnrici [5, p. 6301, we introduce the closely related furictions

(3.50) a,, j(z) := z-,- l t ~ l T r h , ,(z)>


It follows from above that

E'rom the reprcsentation

(see. e.g.. [4] ) one can show that a, ~ ( z ) is closely related to incomplete ganima functions r ( a , z ) and hence to the error function. Modified Bessel functzons of the second kznd K , , ( z ) can be expressed by

By setting w, := rU'"/' we obtain

1 7l (3.10a) J I , ( z ) = [ i ~ , , K , ( z z ) - w ~ , K ~ , ( - ~ ) ] , for / a rgz l < -,

17T 2

1 7l (3.10b) Y ; , ( ~ ) = - - [ W ~ , K , , ( ~ ~ ) + L J ~ , K , , ( - I Z ) ] . for I a r g z l < - .

7l 2

where J , , (z ) and y , ( z ) denote Bessel functions of the first and second kinds. respectively. In particular, with z = x > 0 we have

Hence one can compute both J , , ( x ) and Y;, ( x ) from K,( zx ) , for x > 0 , by (3.11). Some known properties of a, j ( z ) . that are subsequently used. are stated by the following lemma. See [5, Sections 9.12. 10.5 and 12.i3] and [4] for proofs. For converiience we introduce a notation for the following set: (3.12)

1 I ) or ( - ;<$and 1 < a and - - < 0 5 -

2 2

LEMMA 3.1. Let ( a , P ) E A and let

Then: (A) v ( t ) > 0, for 0 < t < cx,

and the moments c k with respect to u ( t ) exist and are given by

ASYMPTOTICS O F STIELTJES CONTINUED FRACTION COEFFICIENTS 175

( B ) a, j (z) zs analytzc for z E S - and adrnrts the c~iynrptotu p o u w 5e71cs expansion

(3.15a) a, ( z ) L ( 2 ) . ns z i x. / argzJ 5 T - E.

where

(C) a,, j(z) zs represen td by the Stzeltjrs t rans form

(D) T h e m i q u e S-fraction

corresyondzng t o L,, 3(z) at z = x ( s ~ e (1 .5 ) and (1 .6) ) conwryes loccdly unrforrnly t o a,, , ( z ) o n S,.

\.Ye come now to the application of Theorem 2.1 to thc fiiiictioils a,, j (z) .

THEOREM 3.2. Lrt ( a . 6) E A ( s ~ e (5' 1 2 ) ) and. for n E W . k f h,, cl'cnofr t h f n t h w e f i r r e n t zn the S - f r a r t ~ o n (3.16b) representntzon of a,, j (z) . T h e n .

( A )

(3.17) b,! 1 lim - = -. 1 - 2

( B ) Let 0 E R s a t z ~ f y 0 < iHI < T . and , fo7 11 E W. I f f g,, ( 2 ) denotr the n / h ( ~ p p r o n ~ n a n t of the S - f rar tmn (3.16b). T h e n fher t errst ronsfnnt5 A3 > 0 and C' > 1 (zndep~nderrt of n and z l but dependent upon 0) such thut , for (111 n E N. IzI > 0 and / arg z / = 0,

PROOF. For all x E R wc define

and let Q : R + R bc defined by

(3.20) (,,(r) = e - ( 2 ( i,

Then clearly Q(-.r) = Q(s) and w ( - s ) = rc,(.r). for .r t R. Thr r fore 111 sllowiiig that Q and w satisfy the conditioris of Tlieorc111 2.1. it s~lff ic~s tc) (oi ls id~r onl? positive valucs of .r. For all .r > 0 o w can verify tlic following. If log .r clmotes thc

d natural logarithm and @ := L,, j(.rL) and a' := -a, tllcn:

tl.1

( 3 . 2 1 ~ ) Q(.r) = ( 2 0 + 2ri + 1) l o g i + .r2 - lop b + log [T ( h + n ) 1% (i + J)]


By (3.15) arid [5. Theorem 11.3fI there exist constants E , € 1 , E L and ~ , j , with 0 < E < E , < E L < E 5 < T such that a:;) (z) has asymptotic series expansions (3.22)

for r r l = 1.2.3. It follows from (3.21) and (3.22) that

.rQ" (1.) x2Q1"(x) (3.24) lim [1 + m] = 2 and lim = 0.

I - - + x 1-+x Q1(x)

From (3.23a) and (3.24) we conclude that thcre exist constants N > 0 and B > 0 such that

BJ- a11 elementary cornputation one can show (from (3.21b) and (3.6)) that

a i d . I T t k = ,-I, then xQ1(.x.) < 6lcr + 4. Therefore there exist constants Al > 0 and > 0 s ~ ~ h that ( 2 . 1 ~ ) holds. Condition (2.3a) holds with c = cr = S = 2. It is rcadil\ seen that the reniaining conditions for Q in Theorem 2.1 hold. Hence by (2.7) n e arrivc at (3.17). which proves (A). (B) is an immediate consequence of (A) ant1 Theorem 1.1 (C). 0

An application of Theorem 3.2 to modified Bessel functions K,,(z) in (3.9) \-ields the following:

COROLLARY 3.3. Let 11 and z satzsfy

Thrv there exists n unique sequence of positave coeficients {a,, (v)):=~ such that


where

Letting cu := 11 and 0 := -u in Theorem 3.2 yields a uiliquc, S-fraction rcpre- sent at ion

where

(3.306) bn(u) - 1 b,,(u) > 0 and 1 - - - if u # 0. r r - r n 2 '

An equivalence transformation 17, Section 2.31 applied to (3.30a) yields

where

(3.31 b) a , , ( u ) = b , , ( u ) / 2 . for 1 1 = 1 , 2 . 3 .

Substitution of b,, ( u ) = 2a,, ( v ) in (3.30b) leads to the assertion (3.28)

EXAMPLE 3.4. Gargantini and He~irici [4] have computed thc m~fficimts n,, ( 0 ) . 2 5 n 5 41. in the S-fraction representation

A few of the computed values a,,(O)/n and 0.25 -a, , (O) /n are given below. rountlctl off to 5 decimal places:

Table 1.

References

. . I.A. Stegun, Handbook of Mathematzcal Funct7oiis uvith F o ~ ~ r n r ~ l n s . Graphs and Mathematzcal Tables. National Bureau of Stantlards. Appl. hlatli. Ser. 55. 1J.S. Govt. Printing Office, Washington, D.C. (1964).

[2] Chihara, T.S.. A n Introductzon t o Orthogonal Polynomzals, h l a t h ~ ~ i ~ a t i r : , a i d Ith Applirations Ser.. Gordon and Breach, Ncw York (1978).

[3] Cizek, J . and E.R. Vrscay, Asymptotic estimation of tlic coefficielits of the col~thueti fraction representing the Binet function, C.R. LIatli. Rep. Acad. Sci. Carlala. \hl. I\'. KO. 1 iAugust 1982), 201- 206.


[ I ] (:;~rgaritil~i. I. ant1 Henrici, P.. X C:ontinuctl fraction algoritlirri for the corriputatiori of higher tr;~lisc.c:r~tI~l~tal fuiictioris in rorr~plcx plane. LIatl~. Corrip. 21 (1967), 18 29.

[.i] Ht.rlriri. P., Applictl and Corriputational Cornplcx Analysis, \'ol. 2, Spccial Functions. Integral Trm~sforlns. Asyrriptotics and Coritinuctl Fractions, Jolm VtTiley ant1 Sons, New York (1977).

[ ( I ] Hvn-itt E. and Strorriherg K.. Real and Ahstract Analysis, Spririgcr Verlag, Kev; York. Inr. ( l M 5 ) .

[7] .Tonc:s. TI-illialri B. and 1V.J. Thron, Continueti Fractions: Analytic Thcory and Applica- t io~is . El~cyclopotlia of hlatllcrnatics arid Its Applicatior~s, 11, A(ldison TVcsley Publishing C'i). . Reatling. h1.A [l!lXO): (listributcd now by Carrlbritlgc University Press. Ncw York.

[X I .JOII(Y. IVilliarri B. a r d TV..J. Throri. Contimicd Fractions in numerical analysis. Applioit Nu- 111i,1icai hIat11. 4 (1988). 1U 290.

[:I] . l i ~ i ~ c ~ . IVilliatri B. anti TV. Van Assche. Asymptotic bellavior of the coliti~iucii fraction co- c+ficic,nts of a class Sticltjcs transforms inclutling the Birict function, Orthogonal F~nlctions. hlouic~nt Tlici~ry and Continued Fractions: Tllcory a1x1 Applications (1V.B. .Tones ant1 A. Sri Ra~lga . ~ t l s . ) . Lcrt111.c Notes in Pure anti Applied hlatl~crnatics. hIarcel Delilicr. Iric. (1988), 257 271.

[ I O] Li~rcwtzc.~~. Lisa arid Haakon Waadcland, Continued Fractions with Applications, Stutlics in C'orr~putational hlatli.. Vol. 3. North-Holland, New York. 1992.

[l I ] Lnhinsky. D.S.. H.N. LIhaskar. E.B. Saff, A proof of Fr.cutl's conjecture for cxpoiicntial w ~ i g l ~ t s , Corlstr. Approx. 1 (1088). 65 83.

[12] I'crroii, O. , D7e Lehm von den Kettmhriich,en. Bard 11. Tcuhricr. Stuttgart (1957). [1:3] Stic1t.jt.s. T.J . , Rccllcrclles sur les fractioris coritiinic, Arm Fac. Sci. Toulouse 8 (18!94). J. 1

122: !1. 1X!)5. 1 17: (<;. \'an Dijk. ctl.). Oeuvres (.'ornpl&tcs-Coll~rt~tl Papers. \'ol. 11. Springer \i,r.lag. Urrlil~. 1!)!):1. pp. 406 570: Eriglisli trans la ti or^ or1 pp. MI!) 7-15.

[I-l] .l'llron. TV..T.. 0 1 1 Parabolic rolivcrgcricc regioris for continuctl fractioris, hIath. Zcitschr. A9 (1!).;8). 1 7 3 182.

[ I 5j \.;111 .\.;s(~li(~, TIhltcr. IIsyinytotzc.s for Orthogonal Polynon~anls. Lecturc Notes in LIathernatics 12(i5. Sprii~gcr \'tlrlag. New York (1987).

[IO] \\-all. H.S.. Anol?yt?c Theoyy of Contanued Fmctzon.~. D. \'an Nostrantl Co., Inr., New York (l!llX).

[17] IVhittakcr. E.T. arid G.N. :Vaston, A Coimse of hlodcrn Analy,s7,s, 4th cti.. University Press, C';rrrlbritlgc (1!)50).

b-111ni1 ~~r id r i is w~ones@euclld. colorado. edu

A Generalization of Van Vleck's Theorem and More on Complex Continued Fractions

A B S I . H , \ C ~ In this work we give allat wc think arc basic an(l useful corl\-c.r- gcnce t l~cory rosults for ronti~iuctl fractions It7(b, , 11) with complrx elvrricmts. \%'c also give a C O I I V L ' ~ ~ I I ~ C thcorein for positi\~c Pcrrorl-Caratl~do(lory contiri- ue(l fractions. All uf the proofs preseritcd are constructive. so t r ~ ~ i ~ c a t i o n c3r.r.or forrnulas follow as a by-product.

1. Introduction

In our opinion, one of the niost useful and basic results in tliv anal>-tic, t licory of cmnt,imied fractions is Van Vleck's colivcrgence theorelr~ for coi~timletl fract ious of th r form K(l /b , , ) wit,ll the h,, lying in a wedge. 111 Section 2 wo gi\rc, a 1 1 1 7 ~

(.onst ructive proof of this t heorein that cont a im a sharp and uscfnl trllil(.itt io l~ t:rror forniula. Also iri Section 2 we give a const,ructivr proof for a II(W. (,o~~\-c~gc.~i(.c> tlicwrem for contirlued fractions K(l /b , , ) whose eleiric~nt s Or , lit, in tlic riglit I i a l f '~ ) l i~~i (~ R(z) > 0 that we feel will provc to be quit,e useful for future rescarc.11 i l l the, sl l t)]c~t of contiilued fractions. \Ye have added some other results in our Vaii V1~c.k f'rirct iow. section that wc hope will he of interest to t,he reader. In Section 3 \w rcsurrwt t li(' Uniform Circle Thcorem for which we recently gave a proof in another public.atiolr and use it to derive a twin convergence region theoreiri for colltinurd fract ic111s I<(l/br,) that we have called t,he lJniform Twin Circle Theorrm. Both of thew results art. generalizations of earlier results that have an intcwstiiig history. \ \ . c s

close Section 3 with a new uniform simple convergence region tlicortm fhr c.ont inuc,tl fractions h 7 ( l / b , , ) in which the boundary of each region is a c,ircle cotit a i i~ i i~g t 1 ~ ' origin in its interior. This rcwilt is provcd with the aid of the twin circ.1~ t licolx~i~i w(, have rneiitio~lcd. In Section 4 we usc our powerful 1966 Uniform Twill Lima(.oi~ Theorcm for conti~iutd fractions K(ci , , / l ) to derivc a new convergcncr tlic~orc~iii for PPC-fract,ions that 11avc played an important role ill invc>stigations oil ~liol~i(mt throry. ortliogonal polyliomials. a d signal processing in thc last two clecatlvs. I\(, believc it is the first rcsult of its t>-pe for these continlied fractions.

\Ve arc, now at a point where w~ think it is appropriate to prcwnt soli~c' ilotntioi~. drfiilitiolis. and results that will be cmployecl in this papcr. Civm t n-o scy11~1~.(5

1991 A1athcmntzc.s Subject Class~f i rc~t~on. Primary ?LIB70: Scco~i(l;rry 4 0 A l 5

180 L. J. LANGE

of complex nunlbers {a , , ) and {b , , ) mith a,, # 0. let

a r1 t,,[LC) := --------- b,, + u: '

and define the coniposition tralisforniations {TI, (w) ) by

where the nuinrrators {A, , ) and the denominators {B , , ) satisfy the three term recurrence relat ions

S \ e define the sequence { f ,, ) by

Then the coutinuetl fraction generated by the sequence t,, is the ordered pair

a,, a1 a2 a,j K(al,/bl,) = K - = - k=lb,, bl + b2 + + . .

f , , , also denoted bl-

is called thc nth approximant of the continued fraction. Convergence of the continued fraction to a number f in the extended complex number system & means convergence of the sequence { f , ,) to f . The A, and B,, satisfy the important determinant formulas

12

A,,-IB,, -A,,B,,I = ( - l ) " n a i # O , n 2 0 !%=I

71 - 1

A,,B,,-2 - A,,-LB,, = (-l)"bll n ab. n 2 1. b=l

The sequence { h , , ) defined by

often plays a major role in convergence proofs for continued fractions. It satisfies the simple recurrence relation

VAN VLECK'S THEOREM AND MORE ON CONTINUED FRACTIONS 181

and, moreover,

If {V,,) is a sequence of subsets of 6 for which

0 E V,,, t I i (K , ) c Ki-I .

then

so that the sets K,, are nested. Hence for m > n, f,,, E h',, and

I f r r l - f n ) I dzam(K,,) =: A,,.

The sequence {A,,) is nonincreasing, so if A,, + 0, then

f := lim f, = n F K n and I f - f,,/ < A,,.

We are now ready t o state the convergence theorem attibuted to Van Vleck [13] for continued fractions

that shall occupy our attention in the next section. A nonconstructivc function- theoretic proof of this result that uses the Stieltjes-Vitali convergencc extension theorem may be found in each of the books on continued fractions by Jones and Thron [3, p. 891 and by Lorentzen and Waadeland [lo. p. 1241. Gragg and IVarner [l] gave a constructive proof of an equivalent theorem in 1983.

THEOREM 1.1 (Van Vleck 1901). Let the elements of K(l/b,,) satisfy

where 6 is an arbitrarily small positive number. Then:

(A) The sequences {f2,) and {f2,,-1) converge to finite values.

(B) The nth approximant f,, is finite and satisfies

(C) K(l/b,) converges if and only zf, in addition,

(D) When convergent, its value f is finite and satisfies

We shall use the following basic theorem.

182 L. J. LANGE

THCOREN 1.2 (Stern (1860)-Stolz (1886)). T h e contmued fractzon K(l/b, , ) t l lo~rq t 5 lf

For a proof of this rvsult s w 13. p. 791.

2. Van Vleck fractions

T:sing t 1 1 ~ termiiiolog?- iiltroduced in [I], we shall call a cont,inued fraction of t 1~ form (1.1) a Val1 Vlcck fract,ion if all R(b,,) > 0 and a st,rict Van Vlrck fract,ion ~vitll wnglv O (0 5 fl < 7r/2) if. ill addition. all ( arg b,, / 5 8. Thus. for strict Van V i c ~ k fractio~is with angle 0 we have

R(b,,) > lb,, / cos 19 > 0.

Hcw is our gciirralization of Van Vlrck's convergcrlce thcorern above for strict Van Vlwk fractions for which wr give a corist,ructivc proof t,liat leads t,o a ncuT. sharp. R I I ~ 11s~fuI truiicat ion Cm-or fiml~lila.

(B ) Thr: seywncp { f , , } , and t h ~ ~ . s thr continuf:d fmct ion K(l/b, , ) , converges to ( I lirtlbt f if tlrrd only z f t h e pr .od~~ct

x.

P := n(l + Ib,.h,, 1 cos0) I.=%


PROOF. IVith V,, , TI, and K,, related as in the Introduction we choose here to set V,, - H. where H := { z : % ( z ) 2 0 ) . Then h',, = T, , (H) and it is not difficult to show. using well known properties of linear fractional transformations, that the diameter A,, of K,, is given by

\Ve set D l , = %(B,, B,,_ 1 ) and recall that h ,, = B,,/B,,_ I . Then

From these incqualities and conditiorls ( 2 . 1 ) it follows that

Now let

Thesc formulas are easily scen to be truc if one uscs tlie fact that

Formulas ( 2 . 5 ) iinply that Ib,, I < m if P corivergrs. since P converges if anti olily if C Id,, / < x. I f I' diverges. then C jb,, I = X. since thr div~rgence of P implies the convcrgr.nce of K ( l / b , , ) and by tlw Sterii-Stolz divt~rgellcr~ theorrm I i ( l / h , , ) would diverge by oscillation if (b,, ( < cc. \Ye haw now essentially taken care of parts (B). (C). and (D). Part (A) follows from seriw niajorizatioli using the filial

184 L. J . LANGE

estimate below.

This coinplet,es our proof of this theorem.

The truncation error estimate (2.4) in Theorem 2.1 is quite practical and considerably sharper in general than the one given in [I] that we shall present soon. What makes this error formula easy to calculate is that the bk are given and the h x satisfy the simple recurrence formula

(2.6) hi, = bi, + l l h k - ~ , ho = m.

To lend some credence to our assertions we present the following special case as a corollar) and then compare estimates.

COROLLARY 2.2. The continued fraction K ( l / n - I ) converges and

1 - - -

2 (2 .7) I f - fn-11 I

n ; = , ( l + l / k ) n + 1 ' PROOF. It is easily verified by induction that

2n+ 1 h2,-l = 1 and h2n = -.

2n Thus: a.fter replacing h k P l by 1 and setting Q = 0 in (2 .4) , we arrive at the result.

Gragg and N7arner [I. p. 11921 gave the following truncation formula for continued fractions K( l /b , , ) whose elements satisfy the conditions of our Theorem 2.1:

1 I f " ' - f n - l ' iicosQlog ( 1 + A C O S Q ~ , ~ b,l)

, m > n ,

where

Upon applying this formula to the continued fraction in Corollary 2.2 we arrive at

where y is Euler's constant. This error estimate is obviously by far not as sharp as the one given by (2 .7) .

The following lemma can be derived from a theorem of Gragg and Warner [l, THEOREM, p. 11931 after replacing a,, by 1, a k by and w by f in the statement of the theorem. The first formula involving p on page 1194 of their proof


can be corrected by replacing the function of p that occurs there by its rrciprocal. But for the sake of completcness, we have rhosen to givc a proof of the lemma here.

L ~ n r n r ~ 2.3. Let %(b,,) > 0 zn the contznued frnctlon K ( l / b , , ) . Then

where

PROOF. In this case we set V,, - HI,, where

H , , : = { z : ! R ( z ) > - % ( b , / 2 ) ) , Ho:={z:!R(z)>O}.

It is easy to see that the condition t,, (Hr , ) C H,,-I is satisfied. With the aid of the determinant formulas in the Introduction we obtain

From this formula and the relation R(h7,) > R(b1,). which can be verified by induction using (2.6). it is not difficult to deduce that

Let us again set Dl, = l /A , so that

where

t := 2 R(hn) /R(bn) > 2.

A simple analysis of the function F ( t ) will show that it has a unique minimum for t > 2 at the point

Straightforward calculations will show that

From the above minimizing process and the fact that

Dl = %(b1)/2 = $112

186 L. J . LANGE

we obtain

Thus. since A,, = l / D l , , we have that

and our lemma follows after some simple algebraic operations.

\\'it11 the aid of this lemrna. we can prove the following results:

THEOREM 2.4. If R(b,,) > 0, then the continued fraction K ( l / b , , ) converges. provided only that the series C fi diverges. If fi diveyqes: then

In particular, zf the sequence {X(b, , ) ) 2.s also rrt,on,otonr. then K ( l / b , , ) conve~qes ~f

PROOF. The proof follows easily from Lemma 2.3 and basic convergence properties of products.

Our next result gives a one-parameter family of continued fractions K ( l / b , , ) illustrating that such continued fractions can converge for complex b,, in the right half plane approaching thc imaginary axis. if a ccrtain simple restriction is placed on the speed of this approach.

THEOREM 2.5. If % ( b , , ) 2 l / ( n + a ) . whew n > -1, then K ( b , , / l ) convwqes. and

PROOF. Under the conditions of this theorern we have that

and

Since the serits C 1 / J ( k + tr)(k + a - 1) diverges by a limit comparison with the harmonic sc,ries, it follows that C 6 = x and therefore that K ( l / b , , ) converges


by Theorem 2.4. After replacing the PI, in the first inequalit) in Theorem 2.4 by their estimates above we arrive at

and our proof is coinplet,e.

It is interesting t o note that for the continued fraction

In comparison. Theorem 2.4 gives the estimate

3. Uniform convergence regions

In this section we present several uniform convergence regloll theorerns for continued fractions K ( l / b , l ) , where additional features of the regions involved are that each satisfies a certain maxiinality property and each is the conlplernent of an open disk containing the origin. IVe refer the reader to [3. p. 64 and p. 991 for formal definitions of the ter~ris best and unzforrn with respect to convergence regions for coiitinued fractions in general that we shall use herc and in the next section. it'<, shall tmploy the following result which we stated and proved in [9]:

THEOREM 3.1 (Uniform Circle Theorem). Let c be a red number, r = d m . und B[r] be the closed unbounded regzon dejmed by

B [ c ] := { z : ) z + 2 c l 2 2 r )

(3.1) = { z : IzI > 2 . exp (Arcsinh ( - c cos (arg 2))) ).

Then B[c] is (L best uniform simple convergence region for continued fractions

188 L. J . LANGE

Furthermore, zf f r l and f denote the n t h approxzmant and lzmzt of (3.21, respectzvely. then, for all n > 0.

From Theorem 3.1 wc can derive the following important uniform twin convergence region result for continued fractions K (lib,, ):

THEOREM 3.2 (Uniform Twin Circle Theorem). Let a and p he real n,umbers such that p > a > 0 and let cu be an arbitrary real number. For n 2 1 let the elements of the continued fraction K z = l ( l / b 7 , ) satisfy the conditions

Then KIT=, ( l / b , , ) convr~yrs unzformly to a finzte lzmzt f and for n > 0

where { f , , } is its sequence of approzim~ants. Furthermore, the twin convergence regions defined by (3.4) are best twin convergence regzons.

PROOF. Wc shall use an idea employed by Hillarn [Z] in his 1962 doctoral thesis but not published elsewhere to show that the uniform convergence of K ( l / b , , ) follows from the Circle Theorem with the aid of an equivalence transformation. We became aware of his transformation technique a few months ago. For any nonzero complex nuinbrr d , the continued fractions KT= ( l / b , , ) and dKF=, ( l / b , , ) are equivalent. whrrc

Now choose

2eP2" d = Jn

Then the conditions (3 .4) for b2,, _ and b2,, become

2a lb~rt-1 - > 2~

Jn 2a > 2~

Jm which are equivalent to the single conditior~


Our assertions concerning the convergence and its speed now follow from the Uni- form Circle Theorem applied to K ( l / & , ) after choosing the parameter c in that theorem in the following way:

In a sequence of two papers. [ll] and [12], Thron had already proved by 1949 that the twin regions (3.4) are twin convergence regions and they are a best pair of such regions. He used the Stieltjes-Vitali covergence extension theorem to establish convergence. Unfortunately, the uniform convergence and truncation error estimates that we have given cannot be obtained by this method. In our next easily applied theorem we introduce a new two-parameter family of simple convergence regions for continued fractions K(l /b , ) in which the boundary of each permissible region is a circle. As we shall see. these regions are also maximal in the sense described in the statement of the theorem.

THEOREM 3.3. Let the elements of the contznued fracfton K,T=l(l/b,,) sa t~s i v the condztzons

(3.6) (b,, - ne"" 2 p , p = Ja2 + 4al sincu) + 4, a > O, cu E EX.

Then K 7 z l (lib,,) converges uniformly to a finite limzt f and for n 2 0

where { f , , ) is its sequence of approxiinants. The region dejined by condztions (3.6) is the largest simple convergence region for h7,~xl (l lb, ,) whose complement is an open disk containing the origin and centered at ae"'.

PROOF. This theorem is really a corollary of the Uniform Twin Circle Theorem. Let b ~ ~ - l E El and b2,, E E2, where

with a > 0 and p = JaL + 4al sin cul + 4 > a. Then by Theorem 3.2 E l . EL is a set of best twin convergence regions for K(l /b , , ) . Furthermore. El 2 EL and El and EL have a boundary point ln common which is either 27 or -22. Thus El is a uniform simple convergence region. If the radius of the boundary circle of El is decreased, then the resulting region ET, say. will contain a segment of the imaginary axis between 22 and -27. Hence, one can construct a periodic continued fraction K ( l / b ) with b chosen from this segment which diverges. Our maxirnality assertion follows from this observation.

190 L. J. LANGE

A positive Perron-Carathkodory continued fraction (PPC-fractzon) is a continued fraction of the form

where

Continued fractions of this type have been extensively investigated in connection with t,he trigonometric moment problem. with Szego polynomials (orthogonal on the unit circle), and with Wiener's linear prediction method used in digital signal processing. Some fundamental information and results on these topics may be found in the sequence of papers [4], [5], [6]. There are convergence results in the literature for the even and odd parts of continued fractions (4.1) that play an important role in moment problems, but not much seems to be known about the convergence nature in general of these continued fract,ions. It is our intention in this section t,o present a convergence theorem which we t,hink sheds more light on the subject. We make the added assumption that

S, ,#O, n = 1 , 2 , 3 , . . . .

Then (4.1) is equivalent to the continued fraction

(4.2)

\Ye shall make use of the following theorem for continued fractions K(a, , 11) which we proved in 1966 [7] and to which we have recently added some luster [8]:

THEOREM 4.1 (Uniform Twin Linlacon Theorem). The cont~nued fractzon K(c f , / l ) converges to n value f E { z : 12-a 5 a ) z f f o r n = l , 2 , 3 , . . . .

where a zs a complex number and cr and a sntlsfy the znequalzty

where

L l (a . a ) := { w : Iw6 - a ( a L - ( # ) I+ aI14 < a(aL - Icr" )

LL(O, 0) := { w : 1%0(1 + 3) - (1 + a ) ( a 2 - 11 + 012)1 - alu'l > ~ ( 1 1 + ( ? I L - CJL) }.

The convergence of K(n, , / l ) is ur~zforrn with respect to the regzons Ll(t r , a) and L Z ( a r a ) an,d these regions are best twin conveTyerLce regions. F ~ ~ r t h m m o ~ . e , zf f,, denotes the n t h approximant of h'(c'','l) or K ( a , , / l ) , then for all n > 1


We are now ready to state our theorern for PPC-fractions.

THEOREM 4.2. Let 0 and a be real numbers .suc.h that

(a( < a < (1 + a / .

Then the continued fraction (4.2) (and hence the continued,fi.c~ction (4.1)) conveyqes ~ ~ n z f o r m l y if for n > 1

PROOF. Set

1 + a - a" 121 2 l + n - a

Furthermore, the t r ~ ~ n ~ a t i o n estimates (4.3) hold. with f,, und f the 71 th approximant and value of (4.21, respectively. nnd c and d given by (4.4).

> + = . C , = / d r 6 , . 4 , := 1,

where the principal square root is t o be chosen Using basic 1n~qli;llities arid straightforward calculatioris. it i5 not difficult to vrrifj that

IcLI,-~ & 7 4 5 a dnd I C ~ , , * 7(1 + 0i)l > m, n = 1.2 , . . .

under the conditions of our theorem. Thus we call apply Theorem 4.1 to the ( . f . K ( c f , / l ) to reap the conclusions we have made 111 Tlleorcm 4.2 mtl drrixe nt the end of our brief argument.

References

[I] W. B. Gragg and D. D. Warner, Two constructive results in continrictl fractions, S1Ail.l .I. Numer. Anal., 20 (1!)X:i), 1187 1197.

[2] K. L. Hillain, Sorm convergence criteria for continued fractions, Doctoral Theszs. U~~ivcrs i ty of Colorado Boulder, 1962.

(31 W. B. Jorics and W. J. Throi~ . Contzwucd Fractzons: Annlytzc Theory and Appltrotzons. Encylopedia of hIathernatics and its Applications . Vol. 11. Ad(liso11 h'eslcy. Reading. blA. 1980.

[4] W. B. Jones. 0. Njdstati, anti W. J . Throii, Continued fractions associatc!d with trigoriornetric arid other strong ~norricrit problcrns, Constr. Appms. 2 (1986), 197 211.

[ii] W. B. Jones, 0. Njastad, ailti W. .l. Thron, Perron Carath/'codory continucti fractioiis. Rn- t lm~a l Approrarrmtzon n7,d Ap2lzcc~tions an AIathmnat~cs and Physzcs, Lecture Notes in hlatll.. \'ol. 1237, 188 206, Springer, Berlin New York, 1987.

192 L. J. LANGE

[ t i ] W. B. Jones. 0. Njastad, and H. Waadeland. Application of Szeg6 polynomials to frequency analysis. S lAh l J. Math. Anal. 25 (1994). 491 -512.

[7] L. J . Langc, On a family of twin Convergence regions for continued fractions, Illznois J . Math. 10 (1966). 97 108.

[8] L. J . Langc. Uniformity and speed of convergence of coniplex continued fractions K ( U ~ ~ / ~ ) , Orthogonal Functzons, Moment Theory, and Continued Fractions, Lecture Notes in Pure and Applied hZathcrnatics. Vol. 199, 275 -326, Narcel Dekker, New York, 1998.

[9] L. J . Lange, Convcrgerice regions with bounded convex con~plements for continued fractions K( l / h , , ) , .J. Comp. Appl. Alnth., t o appear.

[10] L. Lorcntzcn and H. Waadeland, Contznued Fractions with Applicatzons. Studies in Cornpu- tational Mathematics, Vol. 3, North-Holland, New York, 1992.

[ll] W. J. Thron. Twin convergence regions for continued fractions bo + K ( l / b , , ) , Amer. J. Math. 66 (1944) 428 438.

1121 W. .J. Thron, Twin convergence regions for continued fractions bo + K( l /b , , ) , 11, Amer. .J. Alath. 71 (1949) 112-120.

[I31 E. B. Van Vlcck, On the convergence of continued fractions with complex elements, Trans. Amer. Math. Soc., 2 (1901), 215-233.

DEP.\RT~IEXT OF MATHEXIATICS. UNI\.EIISITY OF ~ I I S S O U R I , COLUAIBIA, ~ I ISSOIJRI 65211 E-mail address: j erryQmath.missouri . edu

Contemporary Mathematlcs Volurne 236. 1999

Convergence of Interpolating Laurent Polynomials on an Annulus

Xin Li

ABSTRACT. The convergence of Laurent polynomials that interpolate functions on the boundary of a circular annulus is studied. The points of interpolation are chosen to be uniformly distributed on the two circles of the boundary. It is proved that, for functions analytic on the closure of the annulus, the sequence of interpolating Laurent polynomials displays a type of maximal convergence. It is demonstrated that, in general. there is no equiconvergence between the interpolating Laurent polynomials and the partial sums of the Laurent expansion of the functions analytic in the annulus. Finally, it is shown that the norm of thc projection operator induced by the interpolating Laurent polynomials is of the logarithmic order.

Dedicated t o Professor J e r r y Lunge on his 70th birthday

1. Introduction

Let T ( p ) = { z E @ : lzl = p) , the circle centered at z = 0 and of radius p > 0 in the complex plane @. Throughout this paper. we will assume p2 > pl > 0. We write A(p1. p2) for the open circular annulus { z E @ : pl < lzl < p r ) . A (p l , p r ) for its closure, and dA(p1, p2) for its boundary. For n7, n > 0. let C(nr . n ) denote the set of Laurent polynomials of the form x;=_,,, n ~ zh (an E @). For each positive

integer p, let ui,, = eLT1'p. Define a set of points as follows:

ZA. = p ~ u i h ~ , k = 1% 2, ...% rn,

and. with a fixed a E [0,2r/(n + 1)).

Then {zh); ' , and { z , , + x ) ; ~ ~ are equally spaced points on T ( p l ) and T ( p L ) . respectively. We now consider t he problem of interpolating functions at pomts {zk ) Assume f l and f 2 are functions defined on T ( p l ) and T ( p 2 ) . respectivclv. Let Llrl ,,(z) = L,,, ,,(z; f l , f 2 ) be the Laurent polynomial in C(m,n ) that interpolates f l at { z ~ ) ? ! ~ and f 2 at { z , , , + x ) ; ~ { . that is, L,,, ,, ( 5 ) = C;=-,,, akzh %

L,,, , , (zx) = f ~ ( z ~ ) , k = 1 , 2 . ..., rn.

1991 Mathematics Subject Classification. Primary 41A05, 41A20; Secondary 26C15.

194 XIN LI

This paper is a continuation of the work of Llason [4, 51, Brutman. Vcrtesi and Xu [2], and Pan [6], where the main concern is on t,he estimation of the norni of the operator f H L l l , r , ~ l ( z ; f l . f L ) (with fJ = f l . l r (O l ) . j = 1 . 2 ) for f anall-tic

in A ( p - I , p ) and contiriuous on A ( p p l . p ) ( p > 1 ) . It is shown that the norm is of the logarithnlic order as n -. m. As a consequence, it is easy to verify the following cor~vergencc result: W h e n f i s ana l y t i c o n t h e cmnu lus A ( p 1 . pn), t h e n L , , . , , I ( s ; f 1 : 5 2 ) (iuith ,fJ = f j = 1 , 2 ) converges u n i f o r m l y to f o n A ( p l , p2) u s n -t x. The novelty of this paper is that the general Laurent polynornials L,,l,ll are considered and convergence by such interpolating Laurent polynornials is established under fairly general conditions. The main results include a type of maximal convergence of L,l,,ll as nlrn 4 r ( r > 0) and rn + n + CG. The paper is organized as follows. In Sect,ion 2 , we prove that {L,,,,, , ( z ; f l , f2)) is convergent for z E A ( p 1 , p 2 ) : when m 3 x and n -+ m, as long as f l and f 2 are Rirrnann integrable functions. In Sections 3, we prove the maximal convergence theorem ment,iond above. In Section 4. we study whether t,he equiconvergence phenomenon occurs bet,ween the interpolating Laurent polynomials and the partial sums of the Laurent expansions of functions analytic on the annulus ( p l , p 2 ) , as Walsh proved for int,erpolating polyno~nials and partial slims of the Taylor expansions of functions analytic on th r unit disk. Finally, in Section 5, we give the as>-rnpt,otic order of the norm of L ,,,.,, as nl + o and n + x.

2. Convergence in A(pI , p 2 )

Xote that zv1Lr1 , ,, ( z ) is a polynomial of degree m + n that interpolates z n 7 f l ( z ) and zr" f L ( z ) at { Z ~ } Y = ~ and { z , , ,+~} ; : : , respectively. By Lagrange's interpolation formula, wc have

where

Note that

and

Thus.

CONVERGENCE OF INTERPOLATING LAURENT POLYNOMIALS 195

Now. wc estimate L1 and L2 separately. iVe can write

The factor in front of the sum approaches 1 for all z E A ( p l , p 2 ) as m + x and n + x. We now compare the surn with

Let A l l be an upper bound of 1 f 1 on T ( p l ) and let z E A ( p 1 . p2). Then

which goes to Aero as n + x. Combining the above estimates, we have

which equals

since the last sum is a Rirrrlarin surn of the Riernanri integrable function

Next, we consider L2. We have

since z; = p ~ + ' r 7 ( " + I) ' ' for k = Trr + 1, m + 2, ..., m + n + 1. The factor in front of the sum has limit -1, and, as in the estirriation of L 1 . we can verify that the sum approaches

for all z E A ( p 1 . m ) . as n1 4 x and n + x. Thus, for z E A ( p l . p 2 ) .

lirn L2 = ,,I - x

8 2 ...

196 XIN LI

Therefore, by the estimates of L 1 and L z , we deduce that the limit ,&& L,,, , , ( z ) ,,-x

exists for all z E A ( p 1 , p s ) and equals

Note that this represents a function that is analytic in A ( p l , p L ) . By a similar argument as in estimating L 1 and L 2 . we can show that for every compact set K c A ( p l , p2 ) . there exists a constant CK > 0 such that lL, ,, ( z ) < CK for all z E K. Thus, the sequence { L , , ( 2 ) ) forms a normal family, and so the convergence is uniform on compact subsets of the annulus. We have proved the following result.

THEOREM 2.1. If f l and f 2 are R i e m a n n integrable funct ions o n ' IT(pl) and T(p .>) , respectively, and i f L,,,,,(z) i s the Laurent polynomial that in terpolates f l

and f 2 o n { zk ) ; , ' i l and { z , , + k } ; ; ~ r f , respectively, t h e n

for all z E A ( p l , p 2 ) , and the convergence zs locally u n i f o r m o n a n y compact subset

of A h , p ' 2 ) .

Now. let f be analytic on A(pl , P L ) . If we take f , = f IT(,, > 1 = 1 . 2 . then. according to Theorem 2.1, for z E A ( p l , p2).

lim L,,, ,, ( z ) = -- ,,, - x 8,-x

which equals f ( z ) , by Cauchy's integral formula. We summarize this result in the following.

COROLLARY 2.2. I f f zs analytzc o n a n annulus A ( p 1 , pb) wzth 0 < pi < pl < pl < p;, and zf L,,, ,, ( 2 ) zs t h e Laurent polynomzal f rom C ( m , n) that znterpolates f a t poznts {zL);~+Ir,+', t h e n

lim L r r L , , ( z ) = f ( z ) , 2 E A h , p2). r , , - x r , - x

and t h e convwgenre zs locally unz form o n a n y compact subset of A ( p l , p L )

3. Maximally Convergent Sequences

I11 the previous section, we established the existence of the limit of {L,, ,, ( 2 ) ) as m + x and n + oo for z E A ( p I , p L ) . when the interpolated functions are Riernann integrable. There is no restriction on how nl and n approach oo. In view of Corollary 2.2. we want to know whether L,,, , , ( z ) converges at any points outside A ( p L , p L ) . In this section, by restricting the behavior of the ratio n l m , we demonstrate how to choose subsequences of {L,, , , ( z ) ) that converge to f , a function analytic on A(pl . p 2 ) , throughout the annulus where f is analytic.

LValsh [7, $7.101 considered Laurent polynomials from C ( n , n ) that interpolate at the ( 2 n + l ) s t roots of unity to a function analytic on the unit disk and showed that the partial sun^ containing all the nonnegative power terms of interpolating Laurent polynomials do not have the maximal convergence property in general. Our result will show that , by taking suitable "lengths" (z .e . , the values of m and n) of positive power terms and negative power terms in the interpolating Laurent

CONVERGENCE O F INTERPOLATING LAURENT POLYNOMIALS 197

polynomials. it is possible to achieve some kind of rnaxirnal convergence on the annulus as expressed in the following result.

THEOREM 3.1. Let f be analytzc o n a n annu lus A(p', , p:) w ~ f h 0 < pi < pl < p2 < pi , and let L,,, ,, E C(m,n) b~ the Lnurent p o l y n o n u d that znterpolates f a t poznts {zk);'=t,7'+'. Let r > 0 , and put

61 := max {pi. ( 2 ) pi}

and

T h e n L,,, . (2) converges t o f (z) for each z E A(61, p ~ ) a s rr/m + T cmd m +n 4 x Furthermore, zf f h a s a szngular poznt o n dA(p;. p;) and zj r = log(pi/pl)/ log(p2/p$). t h e n p1 = pi and p2 = pi , and

Proof. Choose p:/ and p i such that

p', < p:' < p1 < p2 < p; < p;

Then. by Hermite's interpolation formula [7, $3.11.

(3.1) crr ' f (<)will 11 ( z )

zn7 f (z) - zn'Llr1 ,, (z) = 4

for z E A(p/i. py), where w,, ,, (2) is as defined in (2.1). Let 111 be an upper bound of f on dA(p:/, py ). Then thc modulus of the integralid divided by lzlrl' is no larger than

A ~ ( ( ( ~ ~ I ( ( Z O ~ ~ + p;71)(l~li1+1 + P;+l) I ( m , n ) := Izln7 ICm - pyll<n+l ~ I + l ~ i ( n + l ) n PL I I C - 2 1 '

We now estimate I(m, n) for ( E dA(p:/, py) and z E A(py, py). (1) Estimation of I ( m , n ) when ICI = py: In this case. we have

Therefore

198 XIN LI

( 2 ) Estirriation of I ( m . n ) when < / = py: 1% havt.

From this. it follows that

Using estimates (3.2) arid ( 3 . 3 ) in (3.1). we can deduce that , for z E A ( p Y , p ; ) .

Lctting p? \ P', and p; /" p/L gives us

( 3 . 4 )

From this, we see t hat, for z E A ( p 1 . p ~ ) , 1 +,

l i~n silp f ( z ) - L,,, ,, ( z ) < 1. r , ,,,-,

t , , . , , .. This proves the first assertion of the theorem. To verify the remaining parts of Theorem 3 . 1 , assume

CONVERGENCE OF INTERPOLATING LAURENT POLYNOMIALS 199

Then

So. j1 = pi alld j2 = Using these and (3.5) in (3.4) . we get. for z E A(P\ ,P>) .

By an application of the maximum modulus principle on A ( p l , p2). this implies

P/1 P', lim sup nisi l j ( z ) - L~~~ ( 2 ) 1 % I lnax {r, -, (T) I }

r , r r , - r 2€T(/ l l (12) I - " ' zI P1 P2 ,, 1 i - x I 1 2

where. by (3.5) , the right hand side is equal to

Assembling cverything together. when r satisfies (3.5) . we have proved the following inequality

Now, to finish the proof of the theorem. we show that the equality must hold in (3.6) if f has a singular point on ~ A ( P ' , , p;) . We need a version of Bernstein's inequality ([3, 51.41 or [7, $4.61) for Laurent polynomials.

L E ~ I M A 3.2. Let 0 < R1 < RL. Then for p,,, ,, E C ( m . T I ) , we h a w

and

Proof. Note that zrr'p,,, , , ( z ) is analytic in @. So, by the maximum modulus principle, we have

nlax Iznlpl l , ( 2 ) I I Illax lzrT1pnl r1 ( z ) 1 , z t T ( R 1 ) - € ~ ( H L )

which iinplies (3.7). The proof of (3.8) is similarly done by appll-ing the maxiinu~n modulus principle

to prr , ,, ( z ) / z U that is analytic in @ \ (0) (including nc).

We now continue our proof of Theorcrn 3.1. U'e need to show that the equality holds in (3.6) . Assume, to the contrary. that, for soinc T E (0.1) arid for all z E x ( p l , P ~ ) . we have

I + , P : liin sup I f ( z ) - L,,, ,, ( z ) 1 < T - ,, / , I , . , 01

Then, in particular,

I +' 4 lim sup I f ( z ) - L,,, I, ,,,1 ( 2 ) 1 < T - .

r n - x PI

200 XIN LI

Thus, by the triangle inequality,

for m large enough. Let R 2 > p2. B y (3.8) in Lemma 4 and (3.9) above. we have

and so.

which equals. by (3.5).

Similarly, from (3.9). by using (3.7) in Lemma 3.2, we can show that , for

Rl E ( 7 4 , pi ),

Therefore, by using the maximum modulus principle, we know that the sequence { L r r , , l r r r L ~ ( ~ ) ) converges geometrically for R1 4 lzl < R 2 . This implies that f has an analytic continuation at least to the annulus A ( R 1 , R 2 ) > a A ( p { , pb), which contradicts the assumption that f has a singular point on d A ( p { , p i ) . So the equality in (3.6) must hold, which completes the proof of Theorem 3.1.

When p1 ~2 = 1> we have the followirig result, which strengthens the implied convergence result of [2, 4, 5, B] as mentioned in Section 1.

COROLLARY 3.3. Let 1 < p < R . I f f i s analyt ic i n A ( R 1 , R ) and zfL,,,, ,-I ( z ) i s the Laurent polynomial that interpolates f a t { z k } f " , w i th pl = pp l and pn = p, t h e n

liin max If ( z ) - L,,,,- 1 (z)l"iL < f I L - x Z I = , , , ~ - ~ - R '

Furthermore, w h e n f has a singular point o n a A ( R p l , R ) , t h e equality holds.

Proof. This follows from an application of Theorem 3.1 (in particular. relation (3.4) in its proof) for m = N and n = N - 1. with pl = p p l , pp = p, p{ = R p l , and pk = R .


4. No Equiconvergence

We continue to use the same assumptions and notations as in Theorem 3.1. Let xr="=_, fAzk be the Laurent expansion of f on A(pi, pi). Set l,,, ,,(z) =

C;=-,,, f L z A , the (m. n)-Laurent polynomial of f . According to [I] and [4]. the operator that maps f to l,, ,, is a projection with minimal norm among all projections from the space of functions analytic in A(pl. p2) and continuous on X(pl, p ~ ) into L(m, n) with respect to the sup-norm. Since the operator mapping f to L,, ,, is also a projection, it is natural to compare L,,, ,, with l,, ,, . By Theorem 3.1 and the well-known property of Laurent expansion, we know that

Is it possible to show that L,,, "(2) - I,,, ,,(z) is convergent in an annulus that is bigger than A(p{, pb)? This is true for the difference of polynomials interpolating at the roots of unity and the partial sums of the Taylor expansion of a function analytic on the unit disk as proved by Walsh, a result usually referred to as the equiconvergence theorem. In this section, we show that no such results hold between the interpolating Laurent polynomials L,,, ,, (z) and the partial sums of the Laurent expansion 1 ,,, , (z). Consider the function

Then f is analytic in A(1/4,4) and p', = 114 and p; = 4. Let's take the points h of interpolation on T(1/2) and T(2). that is, let zk = 4 / 2 and z,,+h = ~ u J ~ ~ w , , ,

k = 1,2, .. .. n. Then the interpolating Laurent polynomial L,, ,, - 1 (2) is given by

and the (n, n - 1)-Laurent polynomial l,, .,, - 1 (2 ) of f is given by

Thus. by a straightforward calculation, we have

It is now immediate to verify that

and

Therefore. there is no equiconvergence between L,,,,, - 1 and l,, ,,-I in any annulus bigger than A(1/4,4).

202 XIN LI

iVe rcrnark that it is possible to split L,, , , I ( z ) into two parts so that one part does equiconverge with Laurent polynomial 1,) ,, ( z ) in A ( 1 / 4 , 8 ) while the other part equiconverges with 1,)- 1 - 1 ( z ) in A ( 1 / 8 , 4 ) . As the above example points out, in general. there is 110 combined equiconvergence in any annulus containing A ( 1 / 4 , 4 ) .

5 . The Norm of L,,,,,

The norm of L,,, ,, is defined by viewing it as an projection from the space of all functions analytic in A ( p l , p2) and continuous on p2) into L(m, n) with respect to the sup-norm. By adapting a construction of Clunie and Mason (see. c.g., [5. pp. 361-3621), we can show that the norm of L m ,, denoted by l L r f l ,,J. is given by

Recall that wnl , , ( z ) = ( zrn - p ; " ) ( z n f l - p ~ f l e " n + l j " ) . Using (2 .2) and (2 .3 ) , we

can writ,e

arid rl+l P ; ~ I ( Z n ~ p;")(Z71+l - p;+l e ~ ( f ~ + ' ) ( k

A, ( Z ) := C I h = l

( n + l ) p ; l ~ I ~ ~ ' ( ~ ; + ~ - p;")(z - z m + ~ ) l '

IVe now estimate X l ( 2 ) and A2 ( z ) when IzI = pl and z / = pz. Let z = ple to for some 6 E [ O , ~ T ] . W e have

and

It follows from these expressions that, as m , n + cx;.


and n + l

&(z) = C 2 p ~ I sin I L A

,=, ( n + l)lpl - p2e'(d+cl-H) I (1 + 0( (2) I T ) }

where the constants in O(. . . ) are independent of z. From a result of Gronwall (see, e.g. . [I, Equations (24) and (25)]), we know that

where y is Euler's constant. Thus,

m -

I sin F 1 C ml sin(: - I;= 1

max Xl(z) = 0 € [ 0 . 2 ~ ] 7i- 7r

Q= x

On the other hand, by a simple calculation, we have

r , ,

So, the maximum value of the above sum is attained asymptotically at 0 = 7rlnl.

Thus. we have, as m, n + cm,

2 max [ X l (z) + A2 (z)] = - log m + 14=p1 7'r

By a very similar argument, we can show that, as m , n + cm.

From these two asymptotic estimates of X I (z)+X2 (z) , we have the following estimate for thr order of the norm of 1 1 Lrl, 1 1 .

THEOREM 5.1. We have

lim IIL,,,.,, I / - - - 2

n1.n-x log max{m, n ) 7i-

If n /m -+ 1, t h e n

for su f i c ien t ly large 71-1 + n .

204 XIN LI

It is known (cf. [I, 41) that the norm of the Laurent projection f (z) H lm ( 2 )

is (4/7r2) log(rn + 72 + 1) + O(1). So, according to Mason [4], L, ,, and I , ,, are both near-minimax approximation within a relative distance of the same order as m, n 4 m. Finally, we remark that (5.1) extends (at least for m + n large enough) Mason's conjecture as verified by Pan [6]. which states

References

[I] Brutman, L., On the polynomial and rational projections in the complex plane, SIAM J. Numer. Anal. 17(1980), 366-372.

[2] Brutman, L., P. Vertesi, and Y. Xu, Interpolation by polynomials in z and z - on an Annulus, IhlA J. Nurner. Anal, lO(1990). 235-241.

[3] Gaier, D., Lectures on Complex Approximation, Birkhauser, Boston, 1987. [4] hlason, J.C., Near-best L , and L1 approximations to analytic functions on two dimensional

regions, zn Multivariate Approximat ion (Ed. Handscomb, D.C.), pp. 115-135, Academic Press, London, 1978.

[5] hlason, J.C., Near-minimax interpolation by a polynomial in z and z p l on a circular annulus, IhlA J. Numer. Anal., 1(1981), 359-367.

[6] Pan, K. , On Mason's conjecture concerning interpolation by polynomials in z and z p l on an annulus, IhlA J. Nurner. Anal., 14(1994), 599-604.

[7] Walsh, J .L., Interpolation and Approximation, Amer. Math. Soc. Coll. Pubs., 20, 1960.

DEPZRTX~ENT OF MATHEXIAIICS. I'NIL'ER~ITY OF CCNIRAL FLORIDA. ORLANDO, FL 32816 E - m a d address xlllpegasus . cc . ucf . edu

Contemporary Mnthernat ics Volume 236. 1999

Convergence criteria for Continued Fractions K(an/l) based on value sets

Lisa Lorentzen

Dedicated to Jerry Lange on hzs 70th birthday

A n S r n ~ c ~ . It has been conjectured by Ruscheweyh and the author that if 01 # V C_ @ is an open, bounded set, and E := {a E @ : a / ( l+V) 2 V) contains at least two elements, then E is a convergence set for continued fractions K(a, , / l ) ; that is, K(a, , / l ) converges if all a,, E E. We proved the convergence if the poles {-h,, ) of

had at least one limit point outside the boundary 6%' of V. In the classical parabola theorem. where V is a halfplane, bounded sub-

sets of E (which correspond to bounded subsets of I/) a r t convergence sets for K(a, , / l ) , regardless of whether {-h,, } approaches dV or not. This was extended by C6rdova in his thesis. In this paper we shall give a shorter proof of C6rdova1s result, and prove convergence in some more cases where {-h,,} approaches the boundary of V. We also extend the results to twin situations. The results are applied to Lange's strip regions.

1. Introduction ,-.

A set V 2 C := C U {m) containing and omitting at least one element. is a simple value set for the continued fraction

(1.2) a ,, S,~(Z) := - E V for all 2 E V and all n E N. l+z

1991 Mathematzcs Subject Classzficatzon. Primary 40A15; Secondary 30D05. Key words and phrases. Convergence. Continued fractions.

205 @ 1989 Amcricar~ R l a t h c ~ r r l ; t t ~ c . a l Sor.ivty

206 LISA LORENTZEN

For short we writc s,, (V) C V. In fact. if F is a function defined on a set A, then F (A) shall denote the set {F (z ) : z E A) in this paper. The approxiinants

nl a2 a,, (1.3) Si,(z) := - = s l o s 2 0 . . . os , , (z ) for n = 1 , 2 , 3 , . . .

1 + i + . . . + G of K(n, , 11) are then contained in the nested sets

if z E V. Here and throughout this paper we use the following notation:

Notation.

0 2 denotes the closure of a set A in e := @ U {cx;). A" denotes its interior. and dA denotes its boundary in e. d(w, A) denotes the euclidean distance, and d, ( t ~ , A) denotes the chordal distance between the set A and a point w E e.

0 A domazn is a connected open set in e. A netghborhood D of a closed set A is a simply connected domain containing A. Circular neighborhoods of a point n E @ are written

A

(1.5) B r ) := {z 6 @ : z - nl < r}, B , (a , r ) := { i E @ : d , ( z , a ) < r}

0 A Jordan domazn is a simply connected domain whose boundary is a simple A

closed curve in C.

0 w is an znner poznt of the simple arc r if it is the image of an inner point in I C R under a continuous univalent mapping cp such that ~ ( 1 ) = l?.

0 The parametrization of the boundary (or parts of the boundary) of a domain V is negatively oriented if V lies to the right when we follow iJV with increasing parameter. Similarly, the paranietrizat,ion is positively orzented if V lies to t,he left when we follow dV with increasing parameter. If it is not, clear which component of e \ dV we refer to, we say that the boundary curve (arc) is negatively or posit,ively orient,ed with respect t,o the domain V.

A

0 If r is an arc in @ represented by z = z( t ) , then

denotes the angle of inclination and the signed curvature of r at the point z(to) + m , if they exist. If z( t ) only has one-sided derivatives of second order a t t = to , we define the corresponding orie-sided curvatures hi(T, i ( to ) ) arid n ( r , i ( t o ) ) . For convenience we also write h i ( r , i ( t o ) ) to mean eit,her onc of them. If I' = a V of a domain V. then (1.6) is based on neg~t~ ive orientation of a V with respect to V.

0 i and 2 denote t,he first and sccond order clerivat,ives of z with respect to t. -

c denotes t,he conlplex conjugate of c E @. F ( x ) = O ( ~ ( . L ) ) as x 4 n means that linlsup,,_, IF(r)/g(x)I < m . F ( x ) = o(g(x)) as s -+ a means that limr,,{b'(z)/g(x)} = 0. a,, - b,, shall mean that a,, - b,, as n + m; that is, lim,, ,,(a,, /hi,) = 1.

CONVERGENCE CRITERIA FOR CONTINUED FRACTIONS 207

It is clear from (1.4) that if 0 E V and

liin diam S,, (V) = 0, r l - x

then K (a, , / 1 ) converges (i.e. the sequence of approximants {S, , (0)) converges in e, which means that either {S , , (O)) converges to a finite number or { l / S , , ( O ) ) converges to 0 ) . Actuallv (1.7) combined with the nesttdness (1.3) implies that { S , , ( z ) ) converges uniformly in V to a constant function S , ( 2 ) = f E V . This kind of unifornlity has not been studied much until recently, but we think it deserves attention for at least two reasons: (i) It is of interest if one uses the approxiinants {S, , (z,, )) instead of the classical ones {S , , ( 0 ) ) . (ii) It is important when the convergence theory for continued fractions is extended to more general functions { s f , ( 2 ) ) A survey of results in this area can be found in [18] . We say that V is a unzforrrr ualue set for K ( a l , / l ) if (1.7) holds. If (1.7) fails. we may still be able to prove that {S,, ( 0 ) ) converges by other arguments.

It is standard to define the element set E correspondmg to V as the set

(1.8) E := { a E @ : a / ( l + V) C V ) .

Then E is a closed set in G ( E may be empty). and V is the value set for mery continued fraction K ( a , , / l ) from E (i.e.. all a,, E E ) . We say that V is a value set for E . (See for instance [ Z l ] . ) The elenlent set E' in (1.8) characterizes thr class of self-mappings (1.2) of V. Note that we allow 0 E E, even though all a,, # 0 in a continued fraction K ( a , , / l ) .

iVe have now set the stage for rnodern convergence tlleorv for contiriued fractions based on value sets. The ideas cvolved around 1940 in works by Scott and Wall [24]. and were irnmediately taken up by others. In particular m7. J. Thron has produced a long list of convergence criteria based on different value sets. some of these in collaboration with his earlier students Mr. B. Jones and L. J . Lange. (See for instance [7] .)

In the following we shall give three examples of such convergence criteria. The first one is interesting because the conditions on V are so mild:

THEOREM 1.1 , ( 2 0 , Thm 1.11. Let V f 0 be an open, boundrd valur set. let E be the correspondzng element set (1.8), and let K ( a , , / l ) be (L contmued frnctzon from E . If E contaens at lrast two elements, and {a , , ) has (L lzmlt poznt @ E*. where

n a E E : -- n a * ~ f 0) with O'V := i ) ~ n ( 1 - all).

l + d * V

then K ( a l , 11) converges to n value f E V and liin,,,, S,, (V) = { f }.

If {a , , ) has all its limit points in E*, we still know that K ( a , , / l ) converges in a number of cases. but we have not (yet?) been able to prove that this happens generally.

Clearly. V is a uniform value set for K ( a , , 11) in Theorem 1.1, and {S , , ( 0 ) ) converges, even though we do not require that 0 E V . The latter is part of a general picture:

208 LISA LORENTZEN

T H E ~ K E A I 1.2. [17, Tlini 1.11. L d V he a bounded value set for K(a , , / l ) such that there e ~ z s t two d ~ s t m c t elements u and 1 ) In V wzth lim,,,, S,,(u) =

11111 ,,+,, S,, ( 1 , ) = j T h e n S,, (0) + f .

If E contains only one point. then there is at most one continued fraction K ( a , , / l ) from E. a periodic one. arid periodic continued fractions are well understood. Hence, the condition in Theorem 1.1 that E contains at least two elements, is reasonable. The strange condition that {a,,) should have a limit point @ EE" is worv. It was conjectured in [20] that this condition can be removed.

The second convergence criterion is a classical result from 1958. For our purpose the following version is sufficient:

THEOREM 1.3. (Thron's Uniform Parabola Theorem [28]). Let oi E R wzth ( 0 1 < 7r/2 ( m d A1 > 0 be fixed constants, and let V be the halfplane

(1.10) V := V, := (z E C : Re(z e- ' " ) > - + cosoi).

T h e n the parabolic region

1s the d e m m t set correspondmg to V,,, and t h ~ r e e n s t s a sequence {d , , ) of posztzve numbers conwrgzng t o 0. dependzng only o n A1 and oi, such that for every contznued fractzon K(n, , / l ) from ,!? := P, n B ( 0 , A I ) .

(1.12) diarrl S,, (V) < d,, for all rr E N.

This theorem has proved veq- uscful in the convergence theory for continued A

fractions. It clearly implies tliat every continued fraction from E converges to a A

finite value. We say that E is a convergence set for continued fractions K(a , / l ) . A

illoreover, V, , is a uniform value set for E. But the convergence is also uniform with respcct t,o K(a , , / l ) from by (1.12). A set E C C with this propert,y is called a m i f o r m conwelgence set for continued fractions K(a , , / l ) .

\Ye can also say that rvery continued fraction K(a , , / l ) from P,, with a bounded element, seclucncr { n , , } converges. We say t,hat an element set E with this property is a c o n v r ~ y e n c e set locc~lly for continued fractions K(a , / l ) . If bounded subsets of E are u n z f o r n ~ convergence set,s. we say t,hat I3 is a locally unzforrn con,vergence set.

Note tliat with the notation from Theorem 1.1. a*V,, = aV,, and P,* = dP, in Thron's Uniform Parabola Theorem, but still we do not need any restrictions on

A

{a , , } c E. The third convergence rcsult is clue to Ant,onio C6rdova Y4venes. and can be

found in his thesis [I]. It gerleralizes parts of Theorem 1.3. C6rdova started with a generalization of Tliron's halfplane V,, in (1.10). He had the following restrictions on his set V:

Y 1: If is i~ ,Jordan doinairl in (C with V = \ (- 1 - V ) , 0 E V, - $ E dV, x E dV and a smooth bo11ntlar~- dV\{m) with continuous curvature. That is, it admits a para~nctrizafion -. = z( t ) , where r;(dV, z ( t ) ) exists and is continuous. (See

( l a . ) Y2: I f { E iJV \ {x), the11 [ - C 2 / ( l + dV)] n dl/' = {C). Y3: If < E dV \ {x) and i3V is negative1~- oriented with respect to V b,y the

pi".~l~letrizi~tiOll z = z( t ) , t 1 1 r ~ 1


We observe that by Y1, i3V = 8" V. Under these conditions C6rdova proved:

THEOREM 1.4. (C6rdova's Convergence Theorem 111). Let V satzsfy YlGY3, and let A f > 0 be gzven. Then the correspondzng element set 2s gzum by

Moreover, E zs a convergence set locally. and V 2s a unzform value set for E n B(0, M ) .

This theorem generalizes the unifornl value set part of Thron's Uniform Para- bola Theorem. This follows since Vc, satisfies Y1-Y3, the parabolic region Pi, in (1.11) can be written Pi, = -C:. where

(1.15) C, := ( a E C : l ~ e (a e L " ) 1 5 cos a ) = [V,, n (-To)] \ {x), 1251.

and C6rdova's element set (1.14) can be written

(1.16) E = -c2 where C := [V n (-V)] \ {x). (See Theorem 3.3.)

It is typical that both Thron's Uniform Parabola Theorem and C6rdova.s Con- vergence Theorem state that E is a convergence set locally. In fact we have the following classical result suggested by Scott and \tTall:

THEOREM 1.5. [24]. .4 convergence set E is always bounded.

This follows directly from the Stern-Stolz Theorem (see for instance [7. Corol- lary 4.20, p. 79]), which says that K(a , , / l ) diverges if

Unfortunately, CBrdova's thesis was never published outside of the University of Wiilzburg. hloreover. his proofs are so involved that it is hard to get a grip on the material. Repeated appeals t o C6rdova to simplify his proofs a i d publish the result have not payed off, and now he has long since turned his time and interest to computer science.

The purpose of this paper is twofold. First of all we want to prove results which extend Theorem 1.1: that is, we want to prove convergence of continued fractions K(a , , / l ) where {a,,) is allowed to approach E*. In particular we shall give a simpler proof of C6rdova's Convergence Theorem. Secondl), we shall go a step further and study twirl value sets (qj, Vl) C e x e for continued fractions K(a , , / l ) from (E l . E L ) ; i.e.. arll-l E El and a ~ , , E Er for all n , where

are the corresponding element sets, and prove a similar result for this situation. It is evident that a l s ~ El and EL are closed sets in C, and that the special case where

= Vl brings us back to the original set-up. Twin regions have been exterisively studied by Thron, Jones and Lauge. (See for instance [6, 9, 10, 12, 25, 271 and the references therein.)

210 LISA LORENTZEN

The extension of C6rdovd's Convergence Theorem to this twin situation is presented III Theorem 2.12 and proved in Section 10. In the special case where q/;, = Vl we are back to C6rdova's situation. except that we do not require that V is connected and d * V = 3V. Besides, the continuity condition on the curvature of i3V in C6rdova.s theorem is weakened. The extension of Theorern 1.1 is given in Theorem 2 4. If we set = Vl in Theorem 2.4, or rather in its Corollary 2.5. we are back to Theorern 1.1, except that we do not require V to be open. (This is no big deal. though. Set Remark R2.1 .I.) In addition we shall prove some convergence results (Theorem 2.7 and 2.10) with weaker conditions than Theoreni 2.4, but they do not guarantee uniformity of the value sets as Theorem 2.4 and Theorem 2.12 do.

As alwam with convergence criteria, it suffices that ( u ~ , , - ~ , a J n ) E ( E l , E L ) from sornc n on. since K ( a , , / l ) converges to a value in @ if and only if its n th tail

A

converges in @. This means in part,icular that we may int,erchange the roles of El and EL! in the convergence criteria. at least as long as we do not claiin that the convergence is uniform with respect to K(n, , 11) frorn ( E l , E2).

Our main results are presented in Section 2. In S~c t~ ion 3 we find the corresponding element sets (1.18) to ( v l . Vl ) in a special case, where one of the conditions is that,

In Section 4 we study some optimality propert,ies for t,his special case. In Section 5 we look closer at our va1il.e sets. Se~t~ions 6, 7. 8, 9 arid 10 contain the proofs of our five classes of convergence results in Section 2. Finally. in Section 11 we apply Corollary 2.9 and t,he generalization of C6rdova3s Convergence Theorem to twin versio~ls of Lange's strip regions [ll].

2. Main results

Let (L5, I/;) be givcn twin value sets with corresponding element sets ( E l . E L ) given by (1.18). IVhcn are ( E l , E L ) convergence sets locally? And when are they locally uniform convergence sets? \Ve shall present some conditions on (V;], Vl) which imply that ( E l , E L ) have such properties. For coiwenicnce we write VJ,, =

MI, VLr,_+l = Vl arid := E l . := EL for all n. We shall also write W,, := @ \ V,, throughout this paper. We shall frequently meet other notations such as (Uo, U1). (i3*vl, i3*Vl), etc for twin sets. Also for these we lct {U,,), {d*V,,), etc, denote the corresponding 2-periodic sequences.

Let K ( a , , / l ) be a continued fraction frorn ( E l , El) . The nested sets (1.4) now take the form

and th r idea is again to prove that

(2.2) diarn S,, ( V , , ) 4 0,


or to prove convergence of K ( a , , / l ) by other means. Evidently, all s,, , and thus also all S,, . are linear fractional transformations. so we may take advantage of their nice mapping properties.

1% denote the approximants of the nth tail (1.19) of K ( n , , / l ) by S ' l n ) ( z ) ; that is.

(2 .3) ' (71) Gl+l a1,+'2 ( I l ,+b 5, ( z ) : = - -- ---- for n.k E N.

1 + 1 + . . . l+z

1. When we choose value sets (I(/;,, V l ) . there is a significant difference betwren the two cases T/;P = 0 and T/;r # 0. If T/;/;P = 8 and El \ ( 0 ) # 0. then also V y = 0 since a / ( l + Vl) C T/;) for all a E E l . This is a rather special situation. and E l # 0 and E 2 # 0 only in very special cases. (See also Proposition 5.5.)

2. If T/;/;P # 8 and E L \ (0) # 8. then also V10 # 0 since a / ( l + v)) C V l for all a E E L . Since a / ( l + V,) C V,-l only if a / ( l + V,") 2 V p , . WE find that ( E l . E L ) are subsets of the clement sets corresponding to (I(:, V,O) if ~ v c disregard the point a = 0 . Hence. we may use (v/;P. VlO) as our value sets instead of (V;,. x). Thls means that it suffices to consider the two situations where (h, V l ) have empty interiors. and where (v). V l ) are open sets.

3. Let v) and Vl be open sets. Since a / ( l + V , ) C VlP1 only if a / ( l +T) C V , ] . we may assume that (V,)' = V,; j = 0.1. In particular this rneans that V, and

\ V , have common boundaries. Moreover, if 5 is simply connected. then

d V , is a Jordan curve in e. We shall see in Lemma 6.1 that if and V l are domains for which the interiors of their complements IVo and Mr1 have one and only one unbounded component each. then we may assume that and VL are simply connected. We may therefore take 6 and Vl to be Jordan domains in such cascs. just as C6rdova did in his convergencc theorem.

- - If cc E r/;)nVl, then both El and E L may be unbounded, which means that ( E l , E L ) can at most be twin convergence sets locally:

PROPOSITION 2.1 Let both E l and EL be unbounded Then ( E l . E L ) are not twzn convergence sets for contzn~~ed fmctzons K ( a , , / l )

This result is well known (folklorr). Its proof is identical to the proof of Theo- rem 1.5.

On the other hand, we have:

PROPOSITION 2.2 . Let (h, V l ) be twzn value sets for K ( a , , / l ) , whew -1 @ - - - A

l()UVl and {n , , ) 2s bounded. Then (V;), V l ) are bounded twzn v d u e sets for. K ( a , , / l ) g w n b?y

(2.4) V, := V, n B(O, R,) for any R, > A1,+l/p,+l.

where AI, := sup ( a l , , + , / and p j := d(-1. q) A

PROOF. IVe have ln2,,+,/(l+lu)l < A I J / p , for all w E V,. so s2,,+ J ( V 1 ) CI 4-1. A A A

and thus s L n f l ( V j ) C s2, ,+,(V7) C VJ-L for 1 = 1 ,2 and all n E N. Vl-I is clearly A A A ,.

nowempty since SL,,+ ( V , ) C V, - 1 , and V, - 1 is clearly # C sincx, V , - 1 C V , - 1 .

212 LISA LORENTZEN

- -

We shall in this paper assume that - 1 # &/;, U Vl . Hence. we are essentially working with bounded value sets. In particular we shall never have cc E Vy for J = 0 or 1. For simple value sets V this is not a severe restriction. It follows from 117, Thin.3.31 that we always may assume that cc is not an interior point in V . This is no longer so for twin value sets. (See for instance 16, 9, 12, 25, 271.) Hence, it really means a restriction in our setting. Still, it is a condition which allows us to set v/;, = V l , which brings us back to the simple value set case. Since cc # V,O, we

can never have - 1 E Vp If 3 ~ i # V , . then - 1 @ 5 - 1 is necessary. The conditions in our first convergence theorem are rather restrictive:

THEOREM 2.3. Let K/;, and Vl be nonempty domazns zn C such that \VJ has one and only one unbounded component each for j = 0,1, and let K O and K 1 be

A

con~pact sets, K o C and K l C V l . Further let E l := { a E C : a / ( l + Vl) CI K O ) # 0 and EL := { a E C : a / ( l + Vu) C. K l ) # 0. Then the followzng holds:

A. There exzsts a sequence {d,,) of posztzve numbers convergzng to 0 , dependzng only on (v), V l ) and ( K O . K 1 ) , such that d z a m ~ , , ( ~ ) 5 d, for all n for every contznued fructmn h 7 ( a , 11) from ( E l , E L ) .

A A

B. ( E l , E ? ) are twin con7iergence .sets for cont,inued fractions K ( a , / l ) . A A

C. If 0 E K. then ( E l , E2) are uniform twin convergence sets for continued fractions K ( a , , / l ) .

In effect the elements of K ( n , , / l ) have to be bounded away from the boundary of ( E l , E L ) in Theorem 2.3. In particular this means that the theorem is empty if E l or E L have empty intrriors. This also means that {a, ,) has to be bounded, so Theorem 2.3 does not contradict Proposition 2.1. Theorem 2.3 is closely related to 15. Thm 4.21. Its proof is given in Section 6.

Our next convergence result is the announced generalization of Theorem 1.1. It has weaker conditions than Theorem 2.3. but it does not claim that the convergence is uniforni with respect to K ( a , , / l ) . We shall use the following generalization of (1 .9) :

for j = 1.2. We shall also use the strange, but standard notation -h,, := S r y l ( m ) for the pole of S,,. Clearly. -h,, # if is bounded, but we may have that d(-h, , . F ) + 0. This may cause problems when we try to prove that diam S,, (F ) + 0. The significance of d*V; and E,* lies in thr fact that if d( -h , , , dV, ,) + 0. tlirn d(-h, , , d*V,,) + 0, and this can only happen if d(a,, . E,:) + 0 under our conditions. (See Leniina 7.2 and Remark R7.1.) IVe get:

THEOREM 2.4. Let (v/;,. V 1 ) be bounded twzn value sets for K ( a , , / l ) such that the correspondzng element sets E l and EL both contazn at least two elements. If {-h2,,+,)E=, has a lzrnzt pomt # 3*4 for J = 0 or J = 1, then K ( a , , / 1 ) converges,

- - and ( F , , V l ) are unzform value sets for K ( a , , / l ) .

- - If - 1 # q/;, U V l in Theorem 2.4, then the boundedness of q/;, U Vl can be replaced

by boundedness of {a,,) to conclude that K (a, , / I ) converges. This is a consequence of Proposition 2.2.


Unfortunately. it may be difficult to obtain information on the limiting behavior of {h,) . The following corollary to Theorem 2.4 gives sufficient conditions on {a , , ) . which of course are easier to check in general:

COROLLARY 2.5. Let ( \ / ; I , Vl) be bounded twin value sets for K ( a , , / I ) such that the corresponding elem,ent sets E l and E2 both contain at least two elenzen,ts. I f { a z 7 L l ) has a limit point a @ ET or {a2, , ) has a linzit point a # E3, then K ( n , , / l ) - - converges, and ( X I , V l ) are uniform value sets for K ( a , , / l ) .

Also here we can require that {a, , ) is bounded and -1 @ 6 U F. instead of requiring that (I(), Vl) are bounded. The connection with Theorem 1.1 is evident. B y setting X I = Vl = V , we get:

COROLLARY 2.6. Let V be a value set for K(a , , / l ) such that the corwspondzr~g element set E contnzns at least two poznts. Let e z the~ {a , , ) he bounded and -1 $ V . or let V be bounded. If {h, ,} has n lzmzt poznt h @ ( - d * V ) , 07 ~f { a , , ) has a l m z t poznt a @ E", then K ( a , , / l ) converges, and V zs a umforrn value set for K ( n , , / 1 ) .

Theorems 2.3 and 2.4 are useful convergence criteria, but the conditions on {a , , ) or {-h,,) relative to the boundaries of ( E l , E L ) or ( F J , V l ) arc anno) ing. in particular in view of Thron's Uniform Parabola Theorem and C6rdova.s Conver- gence Theorem which have no such restriction. It is obvious from Theorem 2.4 that the case where d(-h , , , d*V,,) -- 0 deserves closer attention. We shall put some extra conditions on the closed set a*&) = -1 - d*Vl to prove more. Our first condition is:

C * l : If s , + ~ 0 s , + ~ ( d * V , ) n d*V, # fl for some a , + ~ E EJ:,th: k = 1 .2 , t l m this intersection contains only one point. say s, + l o s J I L (a* \TI ) n a* \: = { C J ) , and = -<;, a , + ~ = - (1 + cj)'; j = 0 , l .

We recognize this as a twin version of C6rdova.s Coridition Y2, and wc call this the Cdrdova Condztzon. The difference is that d * V = d V in C6rdova'5 case, and that C6rdova required the existence of some a E E, depending on (, such that a / ( l + 3*V) n d* V = {C) for every C E d V . It is clear that if C * 1 holds. thcn tlit critical parts of EJ and V , are

Actually, d ( - h,, , a*V,,) + 0 implies that d(-h, , , diV, , ) + 0 when lim sup lh,, I < cc and C* 1 holds. (See Lemma 8.1 and Remark R8.1.2.) It is worth noticing that we

A

always have d T V , = -1 - dTI( / ; ) . and that this is a closed set in @. It is also clear that C * l can be replaced by

e* 1: If s, ( d * V , ) n d*V,- 1 # 8 for sornr a, E E,, then this intcrsection corltains only one point.

e*1 is however more restrictive that C*1. (See Remark R8.1.4.) Hence we shall call e * 1 the strong Cdrdova Condltmn. Evidently. e * 1 holds trivially if a*X/;, has at most one point.

Our second condition depmds on a given A f > 0:

214 LISA LORENTZEN

CS12: There exist positive constants p, q and $ (depending on (h, Vl) and AI) . such that the followiilg hold for every {, E 8t~, with / 5 A l : = 0 , l :

(a): r, ,, := aV, n B ( { , , p ) is a simple and smooth arc. and to every A < E T I , there exists a circular disk Di in @ \ V: with radius q and

with { on its boundar~~: j = 0, 1.

(b): f i r every ( E r, ,, we have cos[arg{(P< - <) /<(1 +<)}I 2 $: J = 0,1. where PC is the center of D,. (That is, with the notation (1.6). p, < + rqe" ('" I .)

For cor~veniencc we also write:

C*2: CT12 holds for ever.y AI > 0.

Condition CT12(n) clearly holds if r3,, has bounded curvature ~ ( r , ,,, <): /r;(rl ,,. <)I 5 l l q . but it may also hold in cases where n( r , ,,, <) is not defined.

To uriderstand Condition CT12(b), we first observe that if Re{(P< - <) /<( I + C)) = 0. and thus the line LC: z = < + ((1 + <)ti t E R. is tangent to dD, at C. then Dc is invariant lirlder the parabolic linear fractional transformation

A

Actually, everv circular disk D< with boundary tangent to L, at { is invariant under

this napp ping. Hence. if C = <, E 8 t ~ ~ . and there exists such a Ei C V , . then the 2 -prriodic continued fraction

A A

from (E,+, , is such that S2,,(D,) = D, c s L r , ( V , ) for all n. and thus (2.2) fails. Hence, the positive < in this condition is related to C6rdova3s Condition Y3. In thc Uniform Parabola Theorem it corresponds to the condition 1 0 1 < 7r/2.

If I/;, and Vl are bounded. then C'2 is equivalent to C,T,2 for sufficiently large AI . 1Jnder Condltion5 C* 1 and C*2 we have convergence of K(a,, 11). even if d(-hr, , i3*V,,) + 0. The following theorem is proved in Section 8:

T H ~ O R E ~ ~ 2.7. Let AI > 0 and let (&/;,, V,) be twin valve sets with v, U Vl C B(0, Al), such that the corresponding element sets (El, E2) both contain at least two elements . If C*l and C1{2 hold. then ( E l , E2) are twin convergence sets for continued fractions K(n,, 11).

Also in this theorern we maj- replace boundedness of ( v , , Vl) by boundedness of { a , , ) if -1 @ U F. This follows from Proposition 2.2:

COROLLARY 2.8. Let ( v l , Vl) be twin valu,e sets such that the corresponding elernrnt .sets ( E l . E2) hoth contain at least t,wo elements . I f -1 $! /;, U and C"1 nnd C*2 hold, then ( E l : E2 ) are twin c o n ~ ~ e n ~ e n c e sets locally for continued fractions K ( a , , / l ) .

For the caw where vl = Vl := V we have dTV .= (2 E d*V : z ' E E and -(1 + 2)' E E). and olir r~ su l t s reduce to:


COROLLARY 2.9. Let V be a value set for h ' (a , , / l ) such that th,e following conditions hold:

( i ) The cor.responding element set E con,tains at least two elemxnts. (12) Condition C * l holds with V;] = Vl = V.

(iii) For every A 1 > 0 there exist positive constants p, q and [ such th,at the following h,old for every E d i V n B ( 0 , M ) :

(a) F, := d V n B ( < o , ~ ) is a simple, smooth arc, and to every ( E F p there is a circu10,r disk DC in \ V0 with radius q and with < on its boundary.

(b) For every < E I', we have cos[arg{(Pc - ()/((I + <)}I > (, where PC zs the center of Di.

If either {a, ,) is bounded and -1 @ V , or if V is bounded, then K ( a , , / l ) con,veyqes.

Theorem 2.7 was inspired by C6rdova's ideas. The conditions in Corollary 2.9 are milder than the ones in Cordova's Convergence Theorem, but we do not get that V is a unzform value set for bounded subsets of E. If we only want to prove convergence of K ( n , , / l ) . WP can do so by even simpler means in the following situation:

THEOREXI 2.10. Let ( q l , Vl) be fwzn value sets for K ( a , , / l ) , where oo E dQl n aV,, 0 @ afi U dVl, -1 @ dVl U dVl and sup / a , / < 1\/f < oo. If there exzst an MI > 0 and a subsequence { n k ) of N such that mk := n k - n k - 1 1s bounded and

-

s!,:'::, (V,,, +,) c BI V,,, for all I ; , where Bn are czrcular dzsks dependzng on K ( a , , / l ) wzth radzz 5 M I , then K ( a , / l ) converges.

The proof of this result is found in Section 9. If we set all mk := m in this theorem and require that every continued fraction K (a,, / I ) with value sets (V;,, Vl) has this inclusion property. we get the following alternative to Theorrm 2.7:

COROLLARY 2.11. Let ('I/;], Vl) be tuizn value sets wzth x E aV;, n dVl. 0 @ dV;) u dVl and -1 @ aV;] U dV1, and let ( E l , Ez) be the correspondzng element spts

A

F ~ ~ r t h e r , let A 1 > 0 and E, := E, n B ( 0 . M ) # @ for 3 = 1 , 2 . If there exzst an M1 > 0 and a v E N, such that to every contmued fructzon K ( a , , / 1 ) from ( E l , E L ) there exzsts a czrcular dask B (dependzng on K ( a , , / l ) ) wzth mdzus 2 A l l , such that

S2,(%) C B C K, then (El, EL) are twzn convergence sets for contznued fructzons h ' (a , , / l ) .

Here we have, without loss of generality, assumed that rrr is even: i.e., m = 2v. It is clear that V,' f 8 in Theorem 2.10, since cc E dV;,naVl implies that 0 E vonV1. and we have required that 0 6 dV;) U dV1, so 0 is an interior point in both and Vl. IVe may therefore. without loss of generality, take and Vl to be open sets.

Neither in Theorem 2.7 nor in Theorem 2.10 was it concluded that (V;), V,) are unzform value sets for the rontinued fractions from (E l , E L ) . To prove and generalize Chrdova's result, we have to establish such uniformity. It turns out that this uniformity follows if the poles {-h , , ) of S,, do not approach dVr, too fast. Actually, the uniformity follows from Lemma 8.3 under our conditions. ~f we can prove that

216 LISA LORENTZEN

(See the proof of Theorem 2.7.) To guarantee that (2.9) holds, we shall impose some extra conditions. Let Rf := {z E R : x > O), and let R denote the class

(2.10) R := {w : R + IRf : ui is continuous at the origin and w(0) = 0)

of functions. Our conditions are then:

CT1: &, and Vl are open sets in @ such that C * l holds, n (-1 - Vl) = 8, O E q/;, and -1 $!I/;,uF.

Ci2: For every A1 > 0 there exist positive constants q and x and an w E R such that the following hold for ever,y <, E dTV, with ( ; I < M : j = 0,1:

(a) T,,, := aV, nB(<,, q) is a simple arc with parametric representations

z = z, (t , w,) := w, + eq,l {t + i&t2 + gJ ( t )) (2.11)

(negative orientation w.r. t. V, )

for every w, E I?, ,. where lg,(t)I < w(t)tL and IgJ(t)l < ui(t)ltl < 5 . (Both g, (t) and the real comtants 7, and bJ depend on w, .)

(b) 2)BJ/ < l /q and 20, - Im (e2'f/w,) > x for every ul, E r, ,.

1. Condition Ci1 is consistent with the set-up in this paper. Neither the condition that &/;, and Vl are open. nor the condition that &,n (- 1 - Vl ) = 0 is particularly restrictive. (See Remark R2.1.2, Proposition 5.4 and Remark R5.1.)

2. Condition CT2 only concerns the parts r J , of dV,. This is sufficient since we can prove that if d(-h,, dV,,) + 0, then d(-h,,, drV,,) + 0. (See Lemma 7.2, Lemma 8.1 and Remark R8.1.2.)

Part (a) demands that these arcs r, , are smooth with well defined curvature. (That z,(t . w,) is twice differentiable at t = 0 follows from the fact that ly, (t)l < d( t ) l t ( . Hence, the curvature exists at z = w,. This holds true for all w, E T, ,.) The condition ui(t)ltl 5 f may be replaced by Reg,(t) > -(1 - E )

for some 0 < E < 1, independent of w, and <, . We do not loose any generality by assuming that the parametrization has

the form (2.11), since the angle of inclirmtion y(dV,, w,) and the curvature ~ (84 , w,) are assumed to exist. This can be seen as follows: If

A

ZJ ( s , wI) := w1 + p l Y ~ {as + bs' + z[jJi!,sL + F, ( s ) ) ; a > 0. b E R.

then the substitution t := as + bs2 brings it over to the form (2.11), at least locally.

3. Straightforward computation shows that


where WJ := @. \ 4. and where we have used the parametrization z = -1 - z J ( - t , w , ) for a( -1 - Mr,). Hence C i 2 ( b ) is equivalent to

- w; (2.13) I ~ ( d V , , w , ) l 5 l / q and K

for every wJ E l-, ,. This is related to C6rdova3s Condition Y 3 if r J , C d iV , .

4. The boundedness conditions on g, ( t ) and /3, are awkward. In the special case where zJ ( t , C J ) is three times differentiable it is no big problem. though. Then it follows that if d V , n B ( 0 , A f ) has finite length and Ct 2 holds for w, := C , , then there exists g > 0 , 2 > 0 and S ( t ) such that C t 2 holds for all w, E r, ,-. This is a consequence of Taylor's formula.

Under these conditions we obtain, with WJ := \ 4: THEOREM 2.12. Let p > 0 be given, let ( I ( ) , V l ) be twin value sets satisfying

C t l and C i 2 , and let the corresponding element sets E l and E 2 both contain at least A

two elements. Let further E, := E , \ { a = -2 . . c E w,o_~ n B ( < , p ) : C E d t V , - l } .

Then ( E l , E2) are twin convergence sets locally and (v), V l ) are uniform value sets for every continued fraction K ( a , , / l ) with bounded element sequence from (El, & ) .

This theorem is proved in Section 10. The restriction to ( E l , E l ) makes sure that c, E 6 - 1 when c, is close to d + ~ , - ~ . This is a technical condition which may be superfluous. Actually, we shall see in Section 3 (Theorem 3.1 in the particular

A

case (3 .5) ) . that if Vl = C \ ( - 1 - 6). as in C6rdova's Convergence Theorem. then -c: E E l only if cJ E V,-l n ( - V , - l ) . So, at least in this case we do not have to restrict the element sets:

A

COROLLARY 2.13. Let (v), Vl) with Vl = C \ ( -1 - K) be twin ,value sets satisfying C t 1 and C i 2 . Let the corresponding element sets E l and E 2 both contain at least two points. Then ( E l , E2) are convergence sets locally for con,tinued fractions K ( a , , / l ) , and ( Q ) , V l ) are uniform value sets for every contmued fraction K ( a , , / l ) with bounded element sequence from ( E l , E 2 ) .

If we add the extra condition rn E 8'V;) = d*l/;) = drl/;,. we call prove more:

COROLLARY 2.14. Let q) C @ with 0 6 d q , . -1 6 d'V;) ( ~ n d cc E d q , , and A

Vl := C \ ( -1 - 6) be Jordan domains such that d t v ) = 8%) and (h, V l ) are twin value sets satisfying e * 1 and C i 2 . Then the corresponding element sets ( E l , E 2 ) are convergen,ce .sets locally given by

(2.14) E , := -c: where C, := V,-1 n (-V,-1) \ {oo) for j = 1.2,

an,d ( K ) , Vl) are uniform value sets for every continued fraction K ( a , , / l ) wzth a bounded element sequence from ( E l , E 2 ) .

If we set V l = v); that is, 'V;) = C \ (-1 - Vo) . then we are back to C6rdova's theorem. except that we do not require that the curvature of the boundary is continuous. Hence, Theorem 2.12 is the generalization of C6rdova3s theorem that we promised in the introduction.

In Section 4 we prove some optimality properties for ( E l , E L ) and ( L o , V l ) in Corollary 2.14. (See Corollaries 4.2 and 4.4 and Theorem 4.4.)

218 LISA LORENTZEN

3. Corresponding element sets

It is in general difficult to describe the element sets ( E l , EL) corr~spoilding to given value sets (&,, Vl). We know by definition that

but what do these sets look like? Lange [ll] has pointed out that it is often easier to describe the sets

which he called c-convergence regzons if every continued fraction from ( E l , EL) converges. This was inspired by Thron's paper [25 ] , where this representation was used for the element set in the Uniform Parabola Theorem. Then El = -Cf and EL = -Cf . IVith our definition, C1 and C2 are larger than necessary. since they are syrnnletric with respect to the origin, but we choose to keep them like this.

The main purpose of this section is to prove (2.14) under appropriate conditions (Theorem 3.3). \Ve shall first make some general observations, though. As already

A

nlentioned. we let If7,, := (C \ V,, for all n . both in this section and throughout this paper. We shall see later (Proposition 5.4) that it is natural to assume that

1 - Vl) = 8 : that is,

( 3 ) & C \ 1 = - 1 - 1 , and thus also Vl C @\(-I-&,) = -l-I$TO.

With this property we easily find:

THEOREM 3.1. Let the twzr~ value sets (&,, Vl) sntzsfy (3.3). Then the correspondmg element sets ( -Cf , - C i ) s a t ~ s f y C1 2 (1 + m1) n (-1 - m ~ ) and C2 c (1 + WO) n (-1 - W o ) .

PROOF. Let c E C1 and assume that c @ 1 - ml. Then w := -1 - c @ ml. That is. ru E V,". However, - cL/ ( l + w) = c 6 -1 - 1/C'I for this particular w. Smre &) C - 1 - I / T I I this contradicts the fact that -cL/( l + Vl) C &I. Hence c E - 1 - 1 1 1 1 . Similarly, -c E -1 - W1. This proves the first inclusion. The sccond onc follows similarly. 0

Actually, it follows from Proposition 5.4 that if (&), Vl) are twin value sets for h ' (a , , / l ) , then so are also (Uo, U1) . where

if Uo # 8. Clearly, U1 C e \ (-1 - Uo) We therefore get:

COROLLARY 3.2. Let ( -Cf . -Cf ) be the element sets correspondzng to the grven value sets (&), Vl). Then C , C q n ( - n , ) w h e r e 0 , := l + f l / I ; U ( - l - V , - ~ ) for J = 1 , 2 .

PROOF. \Ve first note that 0, = 1 + e \ UJ, where U, is given by (3.4) for J = 0,1. Thc rcsult holds trivially if CJ = 8. Lct CJ # 8. Thc rcsult is also trivial

A

if U j = 8. since then also UJ+ 1 = 8. and thus f 2 , = @. If UJ # 8, then (Uo, UI ) arc dl50 twill value wts for (-Cf , C f ) , and the result follows from Theorem 3.1.


In the particular case where we have pi) = -1 - vl, then Theorem 3.1 shows that

It is a remarkable fact that if we add a few conditions, then we can actually obtain equality in (3.5). The idea is due to Cbrdova, [ I ] . I11 the twin situation it takes the form:

THEOREM 3.3. Let @) 2 @ wzth 0 E v), -1 $? and rx, E d v ) . and Vl := C \ ( -1 - V o ) # 8 be Jordan domazns. If

(3.6) (cylav,) n aV, = { < } for every < E BV, \ { m } ; j = 0.1 , -

then the correspondzng element sets (-c;, - C i ) are gwen by CJ = V J - 1 n(-V,- 1 ) \ { c o ) for j = 1,2 .

PROOF. We first observe that also 0 E 6, -1 $? V1 and w E 8 6 under our conditions. In view of Theorem 3.1 and t,he cornments above, it suffices to prove that Po n (-V,,) c C 1 u {m). (The proof for j = 2 follows similarly.) Since l(, and & are Jordan domains. we know that c E C1 if and only if -C"/(l + P I ) 2 V o .

Let C E a@/;,. Then - c 2 / ( 1 + aVl) = <'/a@/;,, which by (3.6) has one and only one point of contact with a@). Since m E aV1 and - c 2 / ( l + 00) = 0 E h, it follows by t,he connectedness of @/;, and Vl that -C2/(1 + V1) c Po. In ot,her words, dl/;, C: C1, and t,hus also -a@/;, C1.

It follows also from (3.6) that (; # (22 if (1 and (2 are two distinct points in aK/;,. Since 8%) is a Jordan curve, it follows that also r := -(dVrl"s a Jordan curve in ,-. A

, ,

C. passing through m. It separates @ into two simply connected components, Po containing the origin and P I . Let vl E Vl be arbitrarily chosen. Since v/;, is simply connected, we must have that either P o / ( l + vl ) c VO or PI / ( 1 + vl ) c Po. Since 0 E @/;,, we must have P o / ( l + v l ) 2 V o . Hence f i / ( l + V1) Vo In other words. f i i El = -C';. The result follows therefore since = -(Vo n ( - V , ] ) ) ' . 0

In the particular case where d*V, = aV,, as in Theorem 3.3, Condition (3.6) corrcsponds to Cbrdova's Condition Y 2 . In fact. (3.6) implies that BTV, = dVJ in

our case, and that E * 1 holds for all a, E E; = E; = B E , We recognize that (2.14) is a consequence of Theorem 3.3, which also gives the formula (1.16) for the eleinent set in C6rdova's Convergence Theorem.

Condition (3.6) is awkward, since it may be hard to check. depending on the shape of B y . If we apply an idea by Lange [ll, Lemmas 4.1. 4.21, we obtain the following result which helps us find sufficient conditions for (3.6):

PROPOSITION 3.4. Let be a smooth arc z = z ( Q ) = r (Q)ed in, C for 0 in some open real interval I of length 5 271.. If r ( 0 ) > 0 and the logarithm,ic derivatiue r l ( Q ) / r ( Q ) is strictly rnonoione in I , then

(3.7) [z(O)"/r] n I' = { z ( Q ) } for every Q E I .

PROOF. Let 00 E I be arbitrarily chosen, and lct K ( 8 0 ) := [z(Oo) ' / r] n T. Clearly, ~ ( 0 0 ) E K ( 0 o ) . We want to prove that z(Oo) is thc only point in K(Oo).

220 LISA LORENTZEN

Assume that z (QI ) E K(Q0) for some Q1 E I. Then there exists a Q2 E I such that z(Q(,)'/z(Q2) = z(Q1). That is.

which means t,hat

Since 6 ln r (0) = r l (Q) / r (0) is strictly monotone, this can only happen if Q1 = QL.

which again gives that 81 = QO. 0

R E ~ I A H K S R3.1.

1. The condition that z = r(Q)e7'; Q E I is a parametrization of r means that r is starshaped with respect to the origin. We do not really need that r is srnooth t o conclude (3.7). The monotonicity of r l (Q)/r(Q) can be replaced by rnonotonicity of the function F ( Q ) := l n r (8 + h ) - ln r (0) for every h > O for all Q E I for which also 0 + h E I .

2. In accordance with Lange [ll]. we say that the arc I?: starshaped with respect to 0, is loyartfhmically convex if it satisfie5 the cond~ t~ons of Proposition 3.4. Lye extend this to arcs r which are starshaped with respect to the origin. for u-h~ch F ( Q ) descr~bed above, is strictly monotone for every fixed h > 0. It follows then that the conclusion of Theorem 3.3 holds if q/;, is a Jordan domain with 0 E V;], -1 @ Vo. x E aq, and logar~thrn~cally convex boundary which is starshaped with respect to -1 and 0. and also Vl := @ \ (-1 V o ) is a Jordan domain.

3. If r is a smooth Jordan curve with O @ r and differentiable parametric representation z = z(f) = x ( t ) + zy(t). t E T i R. then

d 4 t ) - arg z(t) = Irn -, dt 4 t )

and thus r is starshaped with respect to the origin if Im(iZ) > 0 for all t , or I m ( i ~ ) < 0 for all t . Then we can choose Q := arg z( t ) to be a rnonotone mapping. Then r (O( t ) ) := Iz(f) is well defined. and we have

cl rl(Q) - Re(i2) - - l n r ( Q ) = - --- = cot [ a r g ( i ~ ) ]

0% r($) Irn(i2)

Hence. r is logarithmically convex if arg(i2) can be chosen to be a rrionotone mapping into [ -T. O] or into [0, 71. Or. more generally, if for every h > 0

is strictly nlorlotone as a function of t for all t E T for which also t + h E T .

4. If is a srnooth Jordan curve with 0 @ T, and parametric representation z =

z( t ) such that Irn(i2) < 0 and the one-sided limits

i ( t + a t ) - i ( t ) i ( t + a t ) - i ( t ) (3.11) 3+(t) := lini , z P ( t ) : = lirn

-Itdo+ a t A ~ - ~ ~ - a t


exist for all t , then r is logarithmically convex if either

2* ( t ) 4 t ) .*(t) (3.12) Im- < Im --- for all t or Im -- i ( t ) > Im --- for all t .

4 t ) 4 t ) 4 t ) 4 t ) This follows by the following argument. To simplify the notation, we assume that both r ( 8 ) and z ( t ) have well defined second order derivatives. The arguments hold also if these are only one-sided. Straightforward cornputation shows

and

wllere 6( t ) < 0 by (3.8). Hence

d2 sgn 7 {ln ~ ( 0 ) ) = sgn

doL

where sgnz = x/lxl for x E EX \ {O), and sgnO = 0.

4. Optimality and strip regions

The notion strzp regzon was introduced by Lange [ll] to describe a simply connected, closed domain C in C whose complement C \ C has exactly two separate components in @. A typical example is the strip region C, in Thron's Uniform Parabola Theorem, as given in (1 .15) . We recall that PC, := C : is the local

- convergence set in Thron's Uniform Parabola Theorem. Moreover. C, = V,, n ( -V,) \ {m) , where V, is the value set for P,. Hence. the two components D; and D: of @ \ C,, satisfy D; = @. \ V,, and DL = V, \ C',, . where D, contains the negative real axis up to the point -i and D: contains the positive real axis down to the point $.

In this section we shall study three optimality properties for such strip regions. The first one is a generalization of a result from [16] which says that if we include just one arbitrary extra point a # 0 to our set Po, a E C \ P,. then E := P, U {a) is no longer a convergence set locally. The arguments are based on the following ideas:

We say that a 2-periodic continued fraction

is of parabolzc or ellzptzc type. if the corresponding linear fractional transformation

is parabolic or elliptic. In the parabolic case K ( a , , / l ) converges to the one and only fixed point of SL. I11 the elliptic case K(n, , 11) diverges. Now. S2 is parabolic with fixed point < E @ \ (0, - 1 ) if and only if


A

COROLLARY 4.2. Let v, c @ and Vl := C \ (-1 - x) be ,Jordan donmin,s such th,nt C1 and C2 given by (4.4) are strip regions. If (3.6) and Ci2 hold. then ( E l , E2) = (-CP, -C$) are twin convergence sets locally, 7uherea.s (El ~ { t r ) . E2) with a E C\ El are not twin convergence sets locnlly for continued fractions K(n , , / l ) . Moreover. (fi, Vl) are ,value sets for ( E l . &).

The second kind of optimality we shall st,udy is t,he following: IVe say that V is the best limit set corresponding to an element set E if V is a value set for f3. and every 7u E V \ (0. m ) is the value of a convergent continued fraction from E. (We disregard 0 and x in this definition since the elements a,, of K ( a , , / l ) are not allowed to t,ake these values.)

This is a rather recent notion. As far as we know: it was first used in [3]. where the idea of convergence based on {S, , (z , , ) ) was ~ t a r t ~ e d . Earlier. the standard approach had been to concentrat,e on {S,,(O)), which made it natural t,o say that V is t,he best linlue set corresponding t o E if cvery ru E V\ (0. w} is the value of an approximarlt S,, (0) of a continued fraction from E . If V is a Jordan domain with 0 E. V , t,hen V is the best value set, if and only if V is the best limit set.

It was proved alrcadl- in 1942 [13] that the halfplane V,, is the best value set for P,, in (1.11) when cu = 0. and the same was proved for general a: - ~ / 2 < (2 < ~ / 2 , in 1953 [8]. So, indeed. V , is the best limit set corresponding to P,, for these a 's. Actually, V,, is the best limit set for the very thin element set dl',. This remarkable result was proved bj- i'aadeland in 1989 1291. He even proved that every 711 E \ (0, x) is t,he valuo of a 2--periodic continued fraction of parabolic type. from dP,,. where the period begins after the first element. We shall see t,hat, t,hc same holds true undcr more general condit,ions:

THEOREM 4.3. Let v, be a Jordan domain whose boundary is starshaped ~ubth respect to 0 and has 0 @ aq), -1 @ dL(/;, and m E d v , . Let furth,er VL := A

@ \ (-1 - V o ) , and let the element sets (El, E2) corwsponding to (I/;,. Vl) halie - (dVIp1)' \ {co) C E, for j = 1,2. Finally, assume that arg < inm-eases from tr to (2 + T as ( follows the boundary a v , , where a is a jized cons tmt 0 < a < T and ((0) = <(a + T) = m. Then every .ul E Vo \ (0, m) zs the v n l ~ ~ e o,f n 2-p~riodic continued frc~ction K(n, , / l ) ,from (El, E2), uihere

(4.5) a E - ( n ~ , , = -(1 + <)', aL,,+ 1 = -<' for all n , for a C E d h .

This means of course that every w E V o \ (0, m ) is the value of a convergent continued fraction from (E l , E L ) . If Vl = E 2 / ( 1 + v , ) , wc also have that every E \ (0, m ) is tlw value of a convergent continued fraction from ( E L , E l ) . Lye

say that the twin value sets (v). Vl) are t h ~ best twan 1zrn7t sets for ( E l , EL). Note that we do not requirc that every continued fraction from (-C;. -Cj) converges. Actually. Theorem 4.3 shows that if 8%) is also starshaped with respect to -1, then (Vo. V 1 ) are the bcst limit scts for ( d E I , dE2) .

PROOF. of ax,. and 0 I p < 2T.

Let z = C,(fl) := r(0)el' for cr < 0 < n + T be a paramctri~ation let w := pe" E \ (0, x) be arbitrarily chosen. wtlerc ti > 0 and IR want to prove that

224 LISA LORENTZEN

for somt, c E i3VI and ( E dQl . That is.

for soinc Q l . 02 E (0. a + T ) . Let f ( L ) := In r ( r ) . Then (4.6) has a solution ( Q 1 , Q 2 ) = (2.r - y". .r) if and only if the equation

has a solution .I E I := (0. 0 + T ) n ( , T). \Ve first observe that I is a non- empty interval, since (a+,-)/2 < ~ + a / 2 < a+T and ( a + ~ + p ) / 2 > ( a + ~ ) / 2 > a . Next we observe that F ( r ) is a continuous function of x in I. Hence the result follows from the Iiiter~rlediate Value Theorem if we can prove that F ( s ) changes sign in I.

i t + - < k + n + - Assumr first that t u < p < a + T . Then I = (y. +). and p < r ( p ) since 71' E Vo. Hei~ce F ( p ) = In p - f ( 9 ) < 0. If ln p = f ( p ) , then a solution of (4.6) 1s Q 1 = 02 = p. Otherwise, F ( p ) < 0 , and the existence of solutions follows since F ( x ) + +m as x --, [(o + p ) / 2 ] + or x + [ (a + T + p ) / 2 I p .

If p 5 a . then I = ((I,-). and F ( x ) 4 -m as x --, a+ and F ( J ) + +m

as .I 4 [ ( I ? + T + ~ ) / 2 ] ~ . Finall>-, if q > ru + T , then I = (9, a + T ) . and F ( J ) 4 +x ds r. + [(fi + q ) / 2 ] + and F(.r) 4 -x as x + ( a + T ) - . 0

This shows in particular that the closure of the value set,s in Corollary 4.2 have this property if their boundaries are starsllapetl with respect to the origin, since this "double starshapenrss" implies that arg( does not vary too much along the boundaries.

Our third optimalitg propcrty is uniqueness of the value sets. Value sets V for a given element set E are in gcncral not unique. This is for iristarice evident from I'roposition 2.2. However. under the following conditions we have uniqueness:

A

THEOREM 4.4. Let (ql, V l ) be closed best limit s d s for (E l , Eq) such that Vl =

@ \ v/;,o # 0. @. Then (v/;P. V y ) are the unique open twin d u e sets for ( E l . E2) in the senst. that ~f ( I J , l . U , ) clre open tsuin v a h sets for ( E l , & ) , then U, \ (0, m) =

v,:- \ ( 0 % ~ ) fofi,r. j = 0.1 .

P R ~ O F . Lct ( l / l l . C r I ) he open twin value sets for ( E l , E L ) . It is easy to see. [5, Tlmi.4.1]. that Vl l and V I are subsets of the closurcs of every palr of twin value set s for ( E l . E 2 ) . 111 particular, /;,C and C F. and thus also V y \ ( 0 , m) C C', \ { 0 , x) for J = 0.1.

A - A

On the other hand. also (C \ (-1 - lJI ) . @I \ (- 1 - G)) art. opcn twin value set$ for ( E l . E l ) hv Leinma 5.1. Hcnce. V ; \ (0. x) C (e \ ( -1 - C',+ ) ) \ { 0 , x) C

A

(@ \ ( - 1 - V , + , ) ) \ {O. x) = V," \ { 0 , x} for j = 0 , l . This proves the result.

5. The shape of the value sets

Value v t s V C e. or (Vl . V l ) C e x e. tan not be free13 chosen if we want the torre\pontling clcmtmt scts to be nail-empt~ iYe need somc guidelines for how to p ~ t k "good" x aluc sets \Vhat me mean bv "good" in this connect ion may depend oil t ht. appll( d tml T ~ E liabe 111 mind In t lm scc tion we shall corisider the followmg 1s511(",:


Finding alternative value sets for element sets ( E l , E2) corresponding to (\$I. V l ) . This can not always be done (see Theorem 4.4). but we give examples in Lemma 5.1 and Proposition 5.4.

0 The part 8vj of d F j is important. For every C E aiv1 \ (0. - 1, m ) tlierc is a 2-periodic continued fraction of parabolic type from ( E l , EZ) converging to <. This is the reason for the optiinality properties connected with d f l / ; ) . But when do points in a*%/;, belong to d i h ? This is a tricky question if the shape of the corresponding element sets ( E l : E2) is not known. Some necessary coridit,ions are given in [ll]. and some others follow from Propositions 5.5 and 5.6.

0 Some sort of contraction is needed to obtain diam S,,(V,,) -+ 0. In Proposition 5.3 we prove that under mild conditions s,,(V,,) docs not fill V,, 1 . This snilcl form of ~ontract~ion is not necessarily sufficient. though. In Proposition 5.7 wc connect part of our sufficient conditions. C S 2 ( b ) and C i 2 ( h ) , to the shape of

8%). We start by proving two simple, but useful observations:

L E M M A 5.1. Let (qI. x) be twrn value sets for K ( a , , / l ) . Then (e \ ( -1 -

V l ) , \ ( -1 - & I ) ) are also twzn value sets for K ( n , , / l ) .

PROOF. Since s,, (V,,) C V,,- l . where s,, ( 2 ) := a , , / ( l + 2 ) is a bijective mapping A A

of @ onto @, it follows that V,, C s;' (V , , - ] ) which is equivalesit to 5;' ( l i ; , - 1 ) C A

MIrI := @ \ V,, . Since s ; ' ( z ) = -1 - s,,(-1 - z ) , this proves that s,, (V,,) C V,, - 1 is equivalent to s,, ( -1 - b/, , - l) 2 -1 - IV,, . Since both and lI'l colltain at least one element by definition, it follows that also ( -1 - Pi'[. -1 - It\/; ,) arc twin value sets for K ( a , / l ) . 0

LEMMA 5.2. Let (h, Vl ) be twzn value sets for K ( a , , / l ) . If - / I , , E \ I r , , for nn n = no E W, then -h,, E LIT7, for all n > n m If -1 @ V l , then h , , E IT:, f o ~ all n E W . If h,, E 1 + V,,+l for an n = no E W, then h,, E 1 +V,,+1 for d l n > If 0 E h. then h,, E 1 + V,,+l for all n E W .

PROOF. IVe have -h7, = sz l ( -h , , -1) and s; ' ( l l<,-l) C IT;, for all T I . Hence -h,, E LITn for all r t > no if this holds for n = rill. Since h = -1. we have

A A

-h7, E Urn for all n if -1 E IV1. Since (@ \ (-1 - Vl ). C \ (-1 - V ; , ) ) are value sets for K ( n 7 , 11) by Lemma 5.1, and - 1 E - 1 - v/;, iff 0 E I/; , , also the rcmaini~ig statements follow. 0

Our first result says that s , , ( x , ) # T/;,-, under mild conditions. It is ii tn-in version of [20, Thm 3.31, where we have removed the condition that V should be an open set:

PROPOSITION 5 . 3 . Lrt (v,, V l ) be bounded trum lialue sets f o r I i ' (a , , / l ) sut lr that both El and EL each contarns at least truo dements . Then there ezzsfn an E > 0 , dependzng only on (q/;,, V , ) . such that for every n 2 2

- - PROOF. Since vI and Vl are boundcd, we haw that -1 @ \$lUVl. and thus also

E l and EL are bounded. Aforeover. also and Vl contain inore thaii one element each. Without loss of generality we assume that vI and Vl are closed sets. silicc, if

226 LISA LORENTZEN

- - (5.1) holds for ( v l . Vl ). then it holds for (Lo, Vl ). Assume that therc exist n I E E I and n 2 E EL ~ 1 ~ 1 1 that

Then sl(Vl) = v) dsitl s>(V;)) = Vl It is also clear that n l # 0. since otherwise L', = {O}, i.e.. a onc-point set. \vhlch is impossible. In fact. 0 @ I/;,. since V;) =

a1 / ( 1 + Vl ) where Vl i5 hourldcd Hence 0 # E l . Similarly we find that 0 # Vl and 0 # EL. Let 2, E El with 21 f nl , and let Z1 (z) := 31 /(1 + z). Then

and

(5.4) a I v, = s, '(q/;,) 2 s,l 0Z,(Vl) = 1 + ,(1 + Vl). a1

Similarly we must have

Lrt a , /Zl =. r t '". Repeated applications of (5.3) then shows that r "P~'" 'v o C - h for all n . w1m-c V;, is bounded and bounded awa, from 0. Hence r = 1, and we ma\ set 0 < 0 < 27r ~ I I I C C G I # nl . This means that every rotat loll r - "'"v/;, of is tontalilet1 ln for tlus particular 8. Sirnllarl> n2/ZL = e 'y for a p E (0.27r), and c\er\- rotation ep"'-Vl of I; is contained in Vl for this particular p. \Ye have in fact b> (5.3). (5.4) nrld (5.5) that

Lct R : = max{(z/ : z E KI). Thcn @ I C B(0, R ) , and thrre is a point Re"' E 3V) 2 V). I d K := {Re"*'e-"'"; r l E N). Then K C v ) . Since, 0 < 6' < 27r: urr know that K contains at least two points. (K is an infinite set if 8 is irrational.)

A~noilg the points in K: we choose the point (or one of t,he two points) with A A

lnaxiliial distas~ce to 1 . Lct this point be ~ e ' c = 1 + Re'". Then R > R. Let A

furt,her K := (-1 + Rrl"~"'T^: n E W}. According to (5.5) we havc that k c b'). A

Lloreover. sinccl 0 < 9 < 27r. we know that K corlt,ains at least two points. All the /.. A A

poirlt,s in K have the distance R to 1 , but at least one of the points in K has larger distance to th r origin than to -1. This is a contradiction sirlcc ql C E ( 0 , R) . Hencc. S 2 ( v 1 ) is a proper subset of for every pair ( a l . as) E (El , E2) . and thus (5.1) holds for some E > 0 for each such pair ( a l , a z ) . It remains to prove the rxistencr of t,lie uniform E > 0. This follows by a standard compactness argument, since El and E2 arc. closed, bounded sets:

For every pair ( n 1 , n2) E ( E l , E2) we defirlc ~((11. as) := sup{& > 0 : S2(Ql) C \ B(z , E ) for a z E K)). Assulrie that t,lie set I := { ~ ( a l , a?) : ( a l , as) E ( E l . Ez) )

has a clustcr point at 0. T11t:n there exists a seqlience {E,,) from E converging to 0. Since ( E l . E 2 ) is co~npact. thew exists a subsequmce { n k ) of N such that


which is impossible. since S; with a; and aT; also maps Kl into a proper subset of V;), even if one or both of a ; , a$ arc zero. Hcrlce E := inftEE e > 0. Siniilarl) it follows that (5.1) holds for s2 o s 3 , and thus for all n. 0

IVe shall extend yet another result from [20] to the twin situation. It is a uscful result which in some cases allows us to reduce the size of given value sets:

PROPOSITION 5.3. Let (Kl. V,) be twin value sets with -1 @ Vll n V 1 . such that the corresponding element sets El and E2 each ~ o n t a i n ~ s at least ttuo elements. Then also (G \ (-1 - Vl): Vl \ (-1 - v ) ) ) a,re twin value sets for ( E l . E 2 ) .

PROOF. Since (-1 - Ilr lr -1 - W70) are twin value sets for ( E l . E L ) by Lemma 5.1. it follows that Uo := q / ; , \ ( - 1 V 1 ) and UI := Vl\(-1-Kl) satisfy E 1 / ( l + U 1 ) C Uo and E 2 / ( l + Uo) C U1. IVe just have to prove that Uo and U1 are non-empty. As5umc that Uo = 0. Then also 11, = 0 since El /(1 +U1 ) C;,. Hence Vl C -1 - V;, and V;) c -1 - Vi, and thus Vl = -1 - Kl. Since E l / ( l + Vl) C V;), wr then have E l l ( 1 - 1 - h) K) = -1 - Vl. That is. sP1(Q/;,) C Vl for all n E E l . On the other hand s(V1) C v/;, by definition. IVe therefore have s(Vl) = V;, for all n E E l . Similarly. s(V;/;,) = Vl for all n E E L . If m E To, then -1 = s - l ( cc ) E TI for n E El \ (0). Hence -1 @ Vo However. we must have x E -1 - VO = V1, w h ~ h implies that -1 E Vo This is a coiltrddict ion. Hcrlcr K) is bounded, and so is L;. But this contradicts the result in Proposition 5.3. Herice lJO # CI arid UI # 8.

REMARK R5.1. It is clear that thc twin value sets (Uo. C1) mith l', = C; \ (- 1 - % + I ) satisfy U j n (-1 - Ul+l) = 0. as required 111 C'1 It is also t lwr that d*U, = d*Vl and dl UJ = dl Vl, so this operation does not change these iinportdiit parts of dV,. This is consistent with Theorem 4.4. which aavs that if dl V, = dV,, then dV, cannot be changed at all. at least not under the coiiditions o f t hat tlieorern.

The next result shows that if wc want to study the situation wherc 8 V J # 0. then we ma) work with (I(), Vl) with non-empty interiors. or, in view of Remarks R2.1.2 and R'2.1.3, with (V;,. Vl) which are unions of Jordan domains. without loosing much generality. For convenience we introduce the class Cc of curves for a given C E C\{0. -1). It consists of the line LC: 2 = C+C( l+C) t for -x < t < K .

and all circles tangent to L< at C. That is. C, is the class of generalized circles which are invariant under mapping by the parabolic linear fractional transformation

PROPOSITION 5.5. Let (V;,, V1) be t w m value sets with { E df V;, \ (0. -1. x). where conwists of (1 finite union of simple. recfifiable arcs T I : j = 1 . 2 . . . . .I\-. Let ( E l , E 2 ) be the corresponding element sets. Then. uizthouf loss of generalltg (cutting r, i n two parts r f necessary), each r , is part of a C, E Ci. I f there emsts a j E { I . . . . . N } such that C",, is not a circle uiith center at th,e origin or n lzne through the origin, then El \ (0) = {-C2}.

Straightforward computat,ion shows that 0 E Li only if < E R. Similarly. Ci contains a circle with ccnter at the origin only if Re< = -1. It is also easy to see that Ci contains a t most onc such circle, and that if C', is slich a circle. the11

el := -(1 + C)2/(1 + CJ) is not a circle with center at 0. Hrnce. Propositioii 5.5 shows that at, least one of the two elemtilt sets is a one-point sct , disregarding the

228 LISA LORENTZEN

point 0. If both V;, and Vl contain arcs from circles with centers # 0 (or from LC when < @ R). thrn E l = and E L = {- ( 1 + c ) ' ) . again disregarding ( 0 ) . and the bcst limit sets for (El, E L ) are (Vl. V l ) = ({C) . { - I - 0 ) .

PROOF. The esential thing here is that < E dTV;l, and thus -<' E E l and -(I+<)' E E L . So in particular we have - ~ ~ / ( l + V l ) C v) and -(l+<)L/(l+v)) CI V , . Tf7c shall see that this implies the shape of b;,. Let r be a simple arc in 4 1 . If C @ T . then SLr,, (r) C rl) for rn large enough, for some arc ro in V;/;, with < E To, where

S L n L ( 2 ) := T 0 T 0.. . 0 ~ ( 2 ) . - m terms

A

and T is givrn by (5 .6) . (This follows since S L n L ( z ) + < as m 4 CX: for all z E C.) If < E r. thcn < E SL,,, (r) for all rn E N. Hence it suffices to prove that I' C E CC when < E r. U'e therefore assume that < E r for all the arcs r in V;).

Since T (vl) C h. there must exist a t least one rl in v/;, such that S L ~ ~ (r 1) C rl for some m E N. This can only happen if ~ ( r i ) C r l . If we remove r1 from Vo. we still have T(V;] ) C V;,, and so, by repeating this argument. we find that r(I',) C: r3 for all r , C q1. This proves the shape of V;,.

The mapping properties of linear fractional transformations show that Uo := u&,C, is such that S'(U0) = Uo for every pair ( a l . a l ) from ( E l , E l ) with a1a2 # 0.

A

Since ?, .= ( 1 + C) ' / (1 + CJ) C E l / ( I + U O ) . we must have a l / ( l + C , ) =

( - a ~ / ~ ' ) ( - < ~ ) / ( l + ?j) = - ( c I I / C ' ) C ~ C: Ug for all a1 E El \ ( 0 ) . Let 0 # n E El be arbitrarily chosen. If LC C Uo. then we must have

( -a /CL)L< = LC. If 0 @ L C . this means that a = -C2. It remains to consider thr case wherr Ull contains circles.

Let C 1 have the sinallest radius and C x the largest radius among the circles 111 Ul j . Since a / ( l + C,) # a / ( l + C k ) if C, # C k , we must have ( -a /<' )C, = C1 and ( -n /<' )CA = C\ simultaneously for some J , k E { 1 , 2 , . . . , N ) . This can only happen if n l = I<'. If C 1 is the only circle in Uo with smallest radius, then we actual11 need that C J = C l . So. if CI has center # 0. then a = -CL. Similarly. if an! other of the circles in Uo have unique radius and center # 0. then a = - C L , and thus El \ ( 0 ) = {-c').

It remains to consider the case where C , and C, are two circles in Uo with comrnoil radius R . Thelr centers are at <&2R<(1 + < ) / < ( I +<) . If ( -a /<' )C, = C,, then (-a/CL)C:, = C,. and we are done since not both circles can have centers a t the origin. So assume that ( -a /<' )C, = C,. and thus. ( -a /<' )C, = C,. That is,

a - i { < + p i R C ( l + i ) } = < - p i R (('+') where I F 1 = 1 , C' iC(1 + Oi I<(l + 0 I '

simultaneously for both p = 1 and p = -1. Straightforward computation shows that this is impossible. 0

The next result says that dV, can have no corners or cusps at inner points of d T y if # 0. This is in particular helpful if d t V , = d y .

- - PROPOSITION 5.6. Let ( F ] , V l ) with -1 @ V;, U Vl be open twin value sets with

corresponding element sets ( E l , E2) such that the Cdrdova Condition C* 1 holds and El and E 2 each contains c~t least two points. Then d t ~ , has no cusps or corners at clny inner point of $4 \ {m); j = 0 , l .


A

PROOF. Let A 1 > 0 be chosen so large that E;; := E, n B(O. A l ) ; j = 1.2. each

contains a t least t,wo points. Let (GI. GI) be bounded value sets for (El. &); t,hat A

is, for instance, Vl := V, n B(0, AI/p) where p := d(-1. 1/;, U F). (See Propositioii 2.2.) Without loss of generality (see Proposition 5.4 and Rc~nark R5.1) we ma\- - - assume that q/;, n (- 1 - Vl ) = 8.

A

Assume that d v 1 has a cusp or a corner a t an inner point < of die, with angle a E [0, 27r]: a # 7r, where t,lie angle is measured ill thc interior of q/;,. Since

A A

dtVl = -1 - d.bV,, (see the drf i~i t~ion (2.6) of d i 4 ) , arid cl C @ \ (-1 - h). wr then have that 27r - CY is the angle of a cusp or a corner of dV1 at the imer point -1 - C of d ip l . measured in the interior of x. Since < E dtl/;,, we have -C2 E El and - (1 + <)2 E E q .

Assume first that 0 < n < 7r. Then -c2/(1 + dVl) has a cusp or a corner a t C with angle 27r - a since s(z) := -C2/(1 + z) is a conformal napping wit,h s(-1 - <) = (. This is impossible since -<"(I + dVl) C V;] a i d 27r - Q > N .

Assume next that 7r < N 5 27r. Then -(1 + <)'/(1 + /;r) c is inipossihle since -(1 + <)'/(l + d v , ) has a cusp or a corner at -1 - { with angle a .

A

This proves that d t ~ , has no cusps or corners at inmr points. The result, follows therefore since A 1 > 0 can be chosen arbitrarily large. 0

Our last result in this section gives some simpler sufficient conditions for Cow dition C;, 2(b) or Condition C12(b). Thcse conditions will make the chrcking part easier, and the choosing part simpler. Condition Ci2(b) requires that

, - 21P, I < l /q and 2dJ - 1111 2 y for all u; E T I ,, := i3Vl n B(CJ. q )

CT2(b) : 1

where E d'Vl n B(O. Al). OJ := - i r ; ( d ~ , . tr~,), 7, := 7(3V1. u.,),

which of course makes sense only if the curvature k-(dVl, 111,) exist5 for all ut1 E r,, Here q and x are given positlve numbers A5 a cornparison. Coildition CT12(h) requires that the circular disk D< 2 w, with radius q > 0. tangcnt to dll', at (. satisfies

cos[arg{(Pi - ( ) /<(I +<))I 2 [ for all < E r,,, := i3V, n B(< l , p ) .

where <j E dtV, n B(0,AI) and P< is the center of D,.

Here p and [ are some fixed positive numbers. Clearly. such disks exist if the onr- sided curvatures ~ ' ( 8 4 , C ) exist and are absolutely bounded by 1/(2q) for every < E I?, ,. (They may exist even if r;+ ( 8 5 , <) and/or K - ( ~ V , , C) does not exist .) With this notation we have:

PROPOSITION 5.7. Let r be a bounded arc in @, let d > 0 and

and let a ( < ) := arg[<(l + C)] for all < E ro Then the following statements are equivalent:

A l . There exists a c > 0 such that sin[a(<) - y(T, ()I > c for all { E r o .

A2. There elcasts a [ > 0 such that cos > [ for all C E To ulherr

PC w the center of the desk Dc 7n CY12(b) nboue.

If In addztlon r;'(r, () r m t s for 011 < E rO. thrn A1 and AL (LW rqu~ualent to:

230 LISA LORENTZEN

A3. There e r ~ s t s a y > 0 such that r i l ( r ( T ) , C ) - K ? ( T , {) > y for all { E T o , uih r rr

PROOF. Let AI := sup{l(l : { E T ) and let < E To That A 1 H A2 follows since

To prove that A1 ++ A3. we assume for convenience that r;+ (r, {) = ~ ~ ( r . 0 . ( A similar argument proves the inequality for the more general case.) Then r has parametric represcntatiolls z = z ( t ) := < + el" { t + dj tL + g ( t ) ) . C E To. 2 := 1 (T. 5 ) . /3 := - $r;(T, ( ) and g(0) = 0. and straightforurard computation shows that

and d 5 IC(1 + <) 5 M ( l + AI) .

REMARKS R5.2.

1. Proposition 5.7 implies that C;,2(b) holds for the particular choice ( := C, (3'V, if and only if the tangent of dV, at C, has angle of incliniition uniformly srnaller than the ray z = C I ( l + i ; ) t Hence this is a necessary condition for C'T12(b). If dQl = dTq/;,, then it is also sufficient.

2. If E 31 vl and (I = -1 - C o , the11 it fo l lou~ by straightforward computation. using Remark R2.2.3, that

wllerr 7 , := 3.(3V,, G) arid PI := -4 K ( ~ v , , {,) arc supposed to exist. If is

an inner point in iS iL ( , . then [j,, = -PI. and so Condition C T 2 ( 0 ) holds for the choicrs wi := C I . ulJ + 1 : = -1 - {, if and only if

This can only hold if the tangent of dVl at CJ has angle of inclination strictly less than thr ray < , ( I + ( , ) t . uniformly with respect to the points C, E 8'4. This follows from t h ~ (quivale11~r in Proposition 5.7. since


If aTv1 = dV;] and rx, E a v , , as in Corollary 2.14 or in C6rdova's Convergence Theorem, then it follows froin the inequalities above that ~ ( d h , <) + 0 as C + x along dQ1. That is. d v l is almost a straight line when it approches m.

3. Coriditioii Ct2(b) corresponds to C6rdova's Condition Y3 combined with his continuity condition for r;(dV. 5). In Thron's Uniform Parabola Theorem it corresponds to a # *7r/2. In the latter example it is easy to see that for cu = &7r/2. the corresponding element set reduces to the negative r c d axis which definitely is not a convergence set of any kind.

4. The inequalities in Remark R5.2.2 above have an interpretation based on the angles

and

where 2 , (0, (,)I = 1. it follows that

6. Proof of Theorem 2.3

In Theorem 2.3 the twin value sets @/;, and Vl are domains for which the interior A

of their complement s, 117, := C\V, for J = 0, 1 each has one and only one unbounded cornponerit. This fact allows 11s to assume that they actually are simply connected:

LEMMA 6.1. Let ( \ / ; I , Vl) be open and connected twin value sets for h7(a,, /1) such tttat \I/; h,as one and ondy one unbounded component U , for. j = 0.1. T h e n

A - A -

the s imply conn,ected domains (C \ Uo, @ \ U1 ) are also twin value sets for K(tr ,, 11) . A A

PROOF. Let V, := C \ UJ for j = 0.1. and let n = ( L L , , - ~ for some rr E W. Smce i3pl C BV,. we then have n / ( l + i3pl) i /;, GI. Hence it follows by the connectivity of that a / ( l + pl) C GI. In particular the interior of PI is rnapped into the intcrior of GI by the linear fractional transformation a / ( l + z): that is.

A - A -

n / ( l + \ T) C \ &. A similar argument shows that n ~ , / ( I + @ \ Uo) C: @ \ U 1 . 0

Part A of Theorem 2.3 is essentially due to the following lemma:

LEILIMA 6.2. Let the t w i n value sets ( v I . Vl) be sim,ply connected domains, let KO C and K l Vl be compact sets. arrd let K(a , / l ) sat is fy as,,+,,/(l + V,) C qPl for all n E W and j = 1.2. T h e n there exists a sequence {ti,,) of posztiue numbers cowuergkny t o 0, depending only o n (Nl. Vl) and (Ko, K , ) , such that

232 LISA LORENTZEN

PROOF. Since a ~ , , + , / ( 1 + 4) C KIel, it follows that V , # c. According to Riernann's hlapping Theorem. there exist anal) tic, univalent functions pJ in V, such that p , (V , ) = U := { z E C : 12 < 1 ) for J = 0 , l . Then fL,,-l := 90 0 srll-l 0 p,l and fLrl := p1 o S J ~ , o prl map U mto K := p , , (K( , ) U p l ( K l ) , and

It follows then by careful use of Schwarz' Lemma. as done in [14], that FLr, (U) converges to a orleepoint set. uniformly with respect to { f , , ) . Actually. an expression for a uniform bound (2 , ) depending only on K . such that diam F,, ( U ) < d?, for all n is found in [14]. Since SLrl = p t l o FLrl o p ~ . where 90 is fixed. this means that diam S2,, (q,) is bounded by some uniform dl,, + 0 . By continuity we thus have diam S2,, (VO) < d2,,. The nestedness (2.1) implies that also diam SLIl+l (V1) < d ~ , , .

0

Our third lemma shows that even {S , , ( 0 ) ) converges under our conditions. It is related to Theorem 1.2:

L E M M A 6.3 . Let (q,, V l ) be hounded twin ? d u e sets for K ( a , , / l ) , where q/;, contains at least two elements, and assurne that there exist two sequences {u,,) and {v,) such that

(6.2) i n S , ( 7 , ) = i n S ( 7 , ) = f u , , , u,, E V,, , lim inf Iu,, - t i , , I > 0. n - x n - x

Then { h ~ ~ ~ + ~ ) ~ ~ ~ has all zts km7t pomts en 1+V,+,; J = 0.1 , and lim,,,, S,,(O) =

f . We say that K ( a l , 11) converges generally to f when lirn S,, (u , , ) = lim S,, (I , , , ) =

f for two sequences {u, , ) . {u,) with liminf d, (u,,, c ' , , ) > 0. This concept was introduced in [4], where it also was proved that if this holds. then

(6.3) lim S , , ( W , ) = f for every sequence {w,,) with liminf d,(w,,. ~ , ' ( g ) ) > 0 , n - x

where g # f is arbitrarly chosen. Xloreover,

PROOF. Since L$ and thus also Vl contain at least two elements. we can choose a g E -1-Vl such that g # f . Since s l l ( -1 -V , , ) C -l-V,,+l for all n. it follows that 5';' ( g ) E - 1 - Vn+l for all n. Moreover, by (6.4), d,(S;' ( g ) , S;' ( m ) ) + 0 since f E and thus f # x . Hence {hzn+, )~==, has all its limit points in 1 + F+l for j = 0 , l . Moreover, w,, := x can be used in (6.3). and thus S , , ( x ) = S n e l ( 0 ) -+ f .

0

PROOF OF THEOREM 2.3. According to Lemma 6.1 we may without loss of generality assume that and Vl are simply connected. Hence part A follows from Lemma 6.2.

A A

To prove part B. we first observe that also (V;,, V l ) are twin value sets for A A

(El, E2) if K j C C VJ for J = 0.1. Clearly we may choose q/;, and Vl to be bounded, non-empty domains. Hence {S,, ( 0 ) ) converges by Lemma 6.3.


Part C is a trivial consequence of part A if 0 E 6 for both J = 0 and J = 1.

Assume that 0 @ V 1 . We choose bounded value sets (GI. GI) as above. such that A

0 E Vo and supZ,,?, / z / 5 supZ,, IzI =: &I where K := KO U K 1 . From Lemma

5.2 it follows that h,, E 1 + Cn+1 for all n. and thus lh, 5 A1 + 1. In particular h , # m, and we can write

S , , ( z ) = A , , + Q , / ( z + h , ) for n = 1 , 2 . 3 , . . . .

Clearly f E l(o and 1 f - S2, ( 0 ) / < d2,, by the considerations above. Lloreover. if A

z* E ql with lz*I = s ~ p ~ , , : ( ~ z l =: Ah]. then

Hence /Q2,L) 5 ( 2 M + 1 ) (hi + l)d2,, /Ail l . On the other hand

and so

- A A

where 0 < d := d(-1 , & ) < Ih2,, / since -1 @ Vl when K/;, is bounded. 0

7. Proof of Theorem 2.4 and its corollaries

To prove Theorem 2.4 we shall use a couple of lemmas.

L E M M A 7 .1 . Let (q/;,. Vl) be bounded twzn value sets for K ( a , , / l ) wzth n ( -1 -F) = a*&.;,, and assume that the correspondzng element sets El and EL each contazns at least two elements. Then limr,,, S$,,+,(z,) = 0 for all z, E V, \ 3*V, for j = 0 , 1 .

This is a special version of 120, Thm 4.51. The condition n ( - 1 - 1/1) = d*& looks strange, but Proposition 5.4 says that if (GI . PI ) are value sets for K ( a , , / l ) ,

A A

then so are ( % I , Vl) given by V, := V, \ ( -1 - V,* l ) . Since then F n ( -1 -G) =

d * V J . we can always choose our value sets to have this property. To prove Lemma 7.1 we shall use the following two classical results:

THEOREM A. [23. Thm 11. Let {T,)Z==, be a sequence of (non-singular) linear A

fractional transformations, and let D := { z E C : { T , ( z ) ) converges i n e) # 0. Then the limit function T for {T,) satisfies one of the following three possibilities:

(a) T is a linear fractional transformation, i n which case T , ( z ) converges uni- A

formly with respect to the chordal metric in D = @.

(b) T is a function i n D taking on precisely two distinct values, where one of the values is taken at precisely one point. In this case D is either all of e: or D is a two-point set.

(c ) T is a constant function i n D .

234 LISA LORENTZEN

THEOREM B. [2. Thrn 2.31. Let {t,,} be a sequmce of h e a r fructronal trans- format~ons, and define

(7.1) T,, : = t l o t 2 0 . . . o t , , for n = 1 , 2 , 3 . . . .

PROOF OF L E ~ I N A 7.1. Evidently both v) and Vl contain at least two elements in our 5ituation. Assume that {Sir, (zO)} has a limit point cr # 0 for some z(, E V;) \ d * v ) . Let the subsequence { S ~ , , , (zo)) converge to a. {S,, (2)) is a sequence on

A

the compact Rieinann sphere for every z E @. Without loss of generality (taklng subscquencrs) we ma! therefore assume that also Sill I (z,)), SL,,~ ( z ~ ) dnd SL,~, (ZL) Converge. where zo, zl and 2 2 are distinct, and also zl E &. and that hL,, converges to a limit h . (Infinite limits are acceptable.) According to Theorem A we then have the three possibilities:

A

(a) {SLrlA ) converges to a linear fractional trarisfornlation in C. Tlwn tk(z) :=

s,,'~ o S2, ,1 I (z) converges to the identity function by Theorem B. This contradicts the fact that tk(Q)) = ~ 2 , , ~ + l O S ~ , , ~ +2 0 . . . o ~ 2 , , ~ + ~ (XI) C: S Z , , ~ +I (Vl) C v) \ B(zk. E )

for some ZI; E /;, by Proposit,iori 5.3. Hence t,his case can not occur. A

(b) D = @ and {SL)rll ) converges to a function T which takes precisely two values. (We know that D contains at least three points.) Assume first that h # oo. Since wc ma?- write

A , , + Q , , / ( z + h , , ) if h , , : = S ; ' ( x ) # m , (7.2) s , , ( z ) =

A,, + d 2 d if h , , = m.

wc must have AL,,I -' A E K, QL,,, 4 and thus Sir), (2) + 0 for a11 z # -h. Ei-orn Lermna 5.2 we know that -h,, E Pi7,, for all n . On the other hand S i l ( z ) E -1 - V,, + for all z E -1 - Vl. where

Sincc (7.3) shows that + SG~, (z) 4 0 for some z E -1 - &. this means that

{h2rrh ) has all its limit points in 1 +F. This means that - / I E Won (-1 -%) =: H where n H = aV;, n (-1 - v) = d*T/;,. Hence zl) # -h and S ~ , , , ( z o ) -' 0 # a . Also this is a contradiction.

Lct 11 = m. If { f ~ ~ , , ~ ) has a subsequence with elements # x, t,llen we may assume that all h2n1 # x. Since Sar,, ( z o ) E V;, which is bounded. we know by (7.2) that QnrIh = C3(hzrlh) as k + cx;. Hence S;,,, (z) = -Q2rLl / (z + 122rrA )2 + 0 for all z # x.

It remains to consider the case where all h2,,, = x, except possibly a finite nurnber. Then wr may assuinc that all h2,,1 = X. Since Sh,,i (zo) + CY. it follows from (7.2) t,hat Q2r1i -+ CY. Without loss of generality we also assume that A2rri +

A

A. Then S2r11 (z) + A + CYZ in C. which is impossible since it takes more than two values ill C. Herice this case is ruled out also.

(c) { S L , , ~ } convergcs to a constant function in a set D containing zo, z1 and 22. Assume first that hLrIh = x for infinitely many indiccs. Then we may without loss of generality assume that all hLrli = X . So Sir,, (z) = QLTa, + a. whereas SlrIr (z l ) - SjrIi (zo) = d)L7,i (z l - q,) 4 0, which of course is impossible. Hence


h2,1A = cc for a t most a finite riurnber of indices. and we can assume that hL71A # cc for all k . Assume first that h # -20. Then

This is a contradiction. Hence h = -20. However, QLr th = C?(zO + hLIIA )' . and thus -11 E d*I/;/;, by the arguments in case (b). Since zo 6 d* I/;), this case is ruled out. Herice {Si7,(zo)) does not have a limit point cr # 0. In other words. Si, ,(zo) -t 0. Similarly we find that SL7,+l ( z ~ ) -, 0 for zl E Vl \ d*Vl.

The second lernma in this section is connectcd to Lemma 6.3. but its conditions and conclusion are weaker.

L E ~ ~ M A 7.2. Let (K, Vl) be bounded twm value sets for K ( a , , / l ) , and n,sumr that the correspondmg element sets El and EL each contazn, a t lrnst two demrnts Then {h2,,+,) has (dl zts lzmzt poznts zn 1 + V,+I. J = 0 .1

PROOF. By Proposition 5.4 we may without loss of gcnerality assume that Kn(-l -F) = a*vl, since V, := V,\(-1-V7+1) for J = 0 , 1 have t h ~ s property arid d*? = d*V,. Since El and EL each contains at least two eleinents, it follows that so do also 1/;, and Vl. Actually. both I/;, and Vl must contain infinitely nianv elements. It follows froin Lemma 7.1 that lirn,,,, Sill+7 (zJ) = 0 for all z , E V, \ d*V,. This means that if we use the represeritatiori (7.2) for S,, , then Q,, + 0 arid {A,,} has all its limit points in I/;, since S,, (V, , ) C I/;) for all n .

Let h be a limit point for { h ~ , ~ ) , and let IL~ , , , + h . \I'ithout loss of generalitv (taking subsequences) we may assume that ALnA + A. Lct r ~ , , , E -1 - V,, ul0 # .4 be chosen. Then Sly1 ( w ~ ) E - 1 - V,,+l for all n since s; ' (- 1 - V,, ) C - 1 - V,,+ 1

for all l a . On the other hand it follows from (7.3) that

This means that if h ~ , , , # cx, for infinitely many indices k , then (SG~'~ (ui~))+hLrlA ) -) 0 as k + rn through these indices, and thus h E 1 + E.

Assume that hLr lh = m for almost all k . Then we may assume that h2,,, = m

for all k . Hcnce, by (7.3). s;:, ( ~ 0 ) = (wo - ALnh)/Q2,, , -f x which contradicts the fact that SL,,' (IDO) is contained in the bounded set -1 - V,. Hence this cannot occur. Hence h E 1 + K. Similarly we can prove that {hr l l+ l ) has all its limit points in 1 + G. 0

REMARK R 7 . l . It follows from Lcinrna 7.2 that if (l4, V,) are bounded twin value sets for K(a , , / l ) . arid the corresponding elenlent sets contain at least two elements each. thcn {-hZr, +,) has no limit points in y \ d * V,. So, if d(-h,, , dV,,) 4

0, then d(-h,,. a*V,,) -) 0.

236 LISA LORENTZEN

PROOF OF THEOREM 2.4. We first note that by Proposition 5.4, it follows that also (Uo, U 1 ) . given by U , := VJ \ ( -1 - 4+1) for J = 0 .1 . are value - - sets for continued fractions K ( a , , / l ) from ( E l . E L ) . Clearly. their closures (UO, U 1 ) are twin value sets for ( E l , E L ) . Lloreover, Uo and U1 each contains a t least two elements, and a* U, = d* V , .

\Ve shall first prove that q\ d*UJ # 8 for j = 0 , l . Assume that &\ d*Uo = 0. Then = d*UO = & n ( -1 - F). and thus & c ( -1 - F). Then s l (G) C G c -l-~.butwealsohavesl(~)=a~/(l+~)>-a~/~=-l-s~'(Uo)> -1 - K. Hence s l ( F ) = & which contradicts Proposition 5.3. Similarly we find that \ d*U1 # 0.

Let first { - h L r , ) have a limit point -h # d*UO = a * h . Then -h # d f i by Lernma 7.2. Actually, -h # K. Let further {-h2, ,) converge to - h , and let u E & \ ~ * U O . Then SL,, ( u ) + 0 by Lemma 7.1. Since

where lu - ul < diam l/;) < x, it follows that diam S2,,, (K) + 0. and thus that diarn S,, (K) 4 0 by the nestedness of the ST, (V,, ).

This result follows similarly if { - hL , ,+I ) has a limit point # d*Ul = d* V l . That also {S,, ( 0 ) ) converges to the one and only point in lim S,, (K) follows

now by Lemma 6.3. 0

PROOF OF COROLLARY 2.5. Assume that diarnS,,(K) 4 d > 0. Then { - h 2 7 , + 3 ) ~ = = , has all its limit points in a*V, for j = 1,2. This is impossible if {aL7,+,) has a limit point @ EEf for either J = 1 or j = 2. 0


The proof of Theorem 2.7 is based on a number of lemmas. The first ones are rather simple observations:

LEAMA 8.1 . Let (vl. V l ) be twzn value sets for h ' (a , , / l ) such that 0 @ d*l/;) U 3*V1 and C* 1 holds. If lim sup Ih,, 1 < cc and d(-h,, , a*V,,) + 0, then (1 - h,, - h , , , ) + O a n d ( n , , + h f , - , ) + 0 a s n + m.

PROOF. Let ( E l . E L ) be the element sets corresponding to (lo, V l ) , and let lirn sup lh,, / < m and d(-h ,, .8* V,,) 4 0. We first observe that also lirn sup la7, < m since a,, = ( h r , - l ) l ~ , , - ~ . Let { n h ) be a subsequer:ce of N such that a2,,, +

a E E L > - h L f l h + C0 E d * K . -hL ,,,-, - E d * V l , a ~ ~ ~ - ~ + 2 E E l and A

- h L r , , - 2 + <O E a*&). Then

Since 0 # d * q l U 8 * V l , we also have -1 @ d*Q) U a*VL. This means that <, , 6) # A

0 , -1, cc for j = 0.1 and a ,Z # 0. Hence we may write <O = s-I o ? ' ( < o ) , where A

5 ( z ) := n / ( l + z ) and ? ( z ) := Z / ( l + z ) . Hence = ? o s(<o). and it follows from A A A

Condition C* 1 that a = - ( 1 + < O ) L . and 2 = -<:. This means that Cl, = and < I = -1 - C o , which proves the result. 0


1. Lemma 8.1 does not imply that the continued fraction in question is h i t 2- periodic. It is actually an open question whether there exist other continued fractions K(a, , / l) than the limit 2-periodic ones which have d(-h,,, d*V,,) i. 0 under our conditions. It is clear that the 2-periodic continued fractions are the only ones in Lemma 8.1 if dTI/;/;, is a finite set.

3. The particular case where K(a,/ l) is limit %periodic is treated in [19]. There it is shown that every such continued fraction from (E l . Es) converges under mild conditions.

4. Condition C* 1 holds if e* 1 is satisfied. This follows by the following argumer~ts: Let e*1 hold, and let s,+l o s,+l(d*VJ) n d*V, # (2). Then there exist (, E a*%, &+I E V J + ~ and <J+L E a*% w ~ h that (,+I = S ~ + L ( ( , + L ) and (, = s , + ~ ( < , + ~ ) . Since -1 - <,+I = 1

-1 - sy+l(<J) = s,+i(-1 - C,) A E s,+l(V,+i) C V,, it follows that <,+I E a*V,+l. It follows now from C*l that <,+h-l is the o n l ~

point in KA := s,+~(d*V,+k) nd*V,+~-,; k = 1.2. Since = s,+h(<,+h) if and only if -1 - (,+A = s l + ~ (-1 - <,+A-1). we therefore must have that i J + n - 1 = -1 - <,+A, and thus also a,+h = - C 1 / L i h _ , -

- -(I + <I+#.

In the next lemma we use the notation (7.2) for S,,. This lemma. combined with Lemma 5.2, shows that diam S,, (K) 4 0 if and only if G,, := Q,,/d(-h,, . dV,,) + 0 under mild conditions:

LEMMA 8.2. Let (I/;), Vl) be twsn - value sets for K(a,, / l) such that I/;) contams at least two elements, and let -h,,,, E II.',,,, for an no E N. Thrn the followzng holds for n = no:

A . diani S,, (K) = IQ,, ( . diarn (K) s f h,, = K .

B. diam Sr, (V,) < 2lQn Ild(-hr,% 8x1) z f h, # x. C. If cx, E K. then diam s,, (K) > /Q,, I/d(-h,, , dV,,) ~f h, , # x

D. If x @ and Ih,, I 5 Al, then there mists a constant k(l > 0 depending only on Al and V,, such that diam S,, (F) 2 k.0 lQ,, l/d(-h ,, . dV,, ).

E. If d(-h,,, , dV,,,) + 0, then there exist an n 1 2 no and a constar~t ko > 0 such that diarn S,,, (I/,,,) > ko IQ,,, l/d(-h,, . dV,,, ) for every rn > nl for which h711 #

PROOF. Also the set Vl has at least two elements since a2, , /( l + v/;,) C V,. where a ~ , , # 0. hloreover. -h,, E for all n > no since s; ' ( l i~,- ) C L17,, arid -hrl = ~ ; ~ ( - h , l ) . Let n := no.

A. This follows directly from (7.2).

B. Let h,, # x. Whether m E or not, we must have

238 LISA LORENTZEN

where uo E dVn is chosen such that I l l o - W I = sup, E ~ , ,, (11 - W / > $ diam V,, . The result follows therefore since 111, + uo 1 < A1 + M I . where A l l := sup{(u?l : w E Kt} < 33.

E. The result follows from part C for every m such that cx, E K. Assume that m 6 K. arid let P , , w, E 8V;) be chosen such that d(-hL,, . d q l ) = ( h2 , + uj,,J and (w, - v,,l 2 d i a rnG for every u > no/2. Since -hr,, E M',,, for all m 2 no, it follows that

I),, - WI, 4 diam 1 + - as u i x , ( f ~ ~ ~ , # x ) . / h 2 , + i.,. I ' diarn + d(-h2 , . aKi) 2

Hence the result follows for eve11 tn from the first inequality in (8.1) for any kll < f . If x @ the result follows similarly for odd nl. b

To prove Theorem 2.7. we shall consider the products

n ~ ~ , , - ~ / ~ 2 , , and ~ G ~ , , - I / G ~ . , + ~ where G., := ~ Q I I 1

d(-hl,. a v n )

If one (or both) of these products diverges to m , then GL,, 4 0 and/or G>,,+l 4 0. arid so (V). V,) are uniform twin value sets for K ( n l , / l ) by Lemma 8.2 arid the nestedness (2.1). If both the products converge, it turns out that wc still get convergence of K(n , , / l ) . To help us see this. we shall use the eslimate in the followirig lemma, which of course is empty if dtKl = 0:

L ~ n f n l ~ 8.3. Let (I/;), V , ) be twin value sets for K(n r , / l ) with -1 @ U r. such that C*l and C"2 are sc~tisfied, and assume that d(-h,,, ~ I v , , ) + 0 and linlsup Ih,, 1 < x. Then there exist n X > 0 and un n o E N such that

G ,~ -L (8.2) R,, := -- > 1 + Xd(-h,,-y,dV,,) for n > n o . where GA :=

l Q k l GTi d ( - h ~ . ski ) '

PROOF. We shall first prove (8.2) for the case where n is even. Let r,, :=

s2rl-l o sllr for all n. That is.

and thus


Since limsup ( h n ( < ca, we also have that lirnsup )a,,) = lirnsup J(h,, -1)h,,-1) < m. Since all a,, # m, this means that {a , ) is bounded. Hence. by Proposition 2.2 we may without loss of generality assume that Ki and Vl are bounded, say I/;) U Vl C B ( 0 , M ) for some M > 0.

Let p, q and E be positive constants such that C,t12 holds with this M > 0, and let ro I, := d&) n B(C(1.p) and rl, := dV1 n B ( - 1 - 6 , p ) for an arbitrarily chosen Co E d iV j . We shall first see that for no E N large enough and p > 0 small enough, we have:

( 1 ) O @ r o , u ~ l , a n d -1 @ ~ o p u r l p . ( 2 ) h,, E -Wn for all n > 1.

(3) {f~,) ,"=~,, is bounded and bounded away from 0 and 1.

(4) {a,)~=,,(, is bounded and bounded away from 0.

( 5 ) {a,, + h,,-I),"=,,, is bounded and bounded away from 0.

Evidently, d t ~ ; , and dtVl are bounded and closed sets in e. Since a* Vl = - 1 - 3" Vo and -1 @ d*Vo U d* Vl , we also have 0 @ 8*Vn U a*Vl. Hence both -1 and 0 have positive distances to dTQj UdTVl . Since dt V, is closed and bounded, this proves (1). ( 2 ) follows by Lemma 5.2. (3) is a consequence of the fact that d(-h,,, dtV,,) - 0. ( 4 ) follows since Lemma 8.1 implies that (a,, + hk -, ) - 0. Finally (5) follows from (3) and Lemma 8.1, since a,, + h,-1 = h,-l( l - h,,-1) + (a,, + f ~ : ~ ~ ) .

Let no be so large and p > 0 be so small that (1)- (5) hold, and let 2n > no + 2. Straightforward computation shows that

Since T; ( F ) ) > & j , it follows that

Let z,, E d h be chosen such that

Then d ( - h ' ~ , ~ - 2 , d&j) 5 )h272-2 + z n ) and the lemma follows for even indices if we can prove that

Since by (8 .4) ,

and h2,,- 1 = 1 + a2,,-1 /h2,1-2. it suffices to prove that

where

240 LISA LORENTZEN

The result follows therefore if we can prove that - z,,)$=~=,,, is bounded, and that there exists a constant p > 0 such that

for all n 2 (nil + 2 ) / 2 large enough. We shall first prove that (h2,-2 + z,,) 4 0. Since Vo C r; ' (K/; , ) , we have by

(8.6) that

where by (8.4)

Since { n , , ) and {a2, ,_1 + h2n-2)~=rl , , i2 are bounded and bounded away from 0 (see Properties (4) and (5) above), this proves that (hsrLP2 + z,,) --t 0. From this it follows that if no is chosen large enough, then z,, has the following properties for all n > n o / 2 :

is bounded. (6) (~271-1 - 2 7 1 171=no/2

( 7 ) z,, E To ,. (See Condition C;[2(a ) . )

( 8 ) z,, is uniquely determined.

(9) { ~ , ; ~ ( z , , ) ) ~ = ~ ~ ~ , ~ ~ is bounded and bounded away from 0 and -1.

It remains to prove (8.8). Let no be so large that these properties (6)- (9) hold. Let further no be so large

and q > 0 in C,:,2 be chosen so small that lasn-l - P,,, I > 2q for all n > n 0 / 2 . This is always possible since by Lemma 8.1, (al,,-l - z,) ( - h & - 2 + h ~ , ~ - l ) =

h21 f -L(1 - h 2 r l - ~ ) which is bounded away from 0. (See Property (3) above.) Since T o , is smooth. we know that the vector from z,, to P,,, is normal to d f i at

z,, . Sincr 7;' is a conformal mapping, this means that the vector from 7;' (z,) to A

PI, is normal to r;'(aK) at ~ ; ' ( z , ) , where F,, is the center of the disk r ; ' ( D Z n ) . (r , ; ' (D_, , ) is a circular disk with radius ja2,-la2,1q/(la~n-1 - PZ,, 1' - q 2 ) since ad,, - I $! Dz, .) On the other hand. it follows from (8.6) that the vector from 7;' (5) to ~ , ' ( - h ~ , ~ - ~ ) is normal to r;'(aV/;,) at r; '(z , , ) . Since both r;'(-h2,-2) @ Vo and PI, @ Vo, this means that we can choose the arguments such that

(8.10) arg ( ~ ; l ( - h ~ ~ - ~ ) - Tll ( & ) ) = arg(F, - r l l ( z n ) )

Let us write a,, := - c i for all n , where we choose c, to be the square root of -a, which is closest to h , , 1 . Since 7,; ' is a conformal mapping, we find that


Combined with (8.10) this gives

C2n- 1 C2n = arg(Pzn - z,) + 2 arg

a2,-1 - 2,

a2n- 1 a2n + arg (1 + azn + ~ ~ ~ ( - h 2 , - 2 ) ) ( 1 + (3277 + ~ l ~ ( z , , ) ) '

From Lemma 8.1 it follows that (1 - h,-' - h,) + 0 and (-c: + hi-,) + 0. We have found that (h2,-2 + z,) + 0 and (r;'(z,) + h2,-2) -+ 0 under our conditions. Since -h,, and thus also z,, is bounded and bounded away from 0 and -1, this means that

so that a2,-laz,/{(l+ a2, + 7;' (-hzn-2))(1 + az, + r;'(z,))) -+ 1. For a given E > 0 there therefore exists an n, > no such that we can choose the arguments so that both

(8.12) arg c 2 n 1 c 2 n 1 < a r g a2n- 1 a2n

a2n- 1 - Zn (1 + + 7;' (2)) (1 + a2, + r i l (2,))

for all n > n,/2. It follows from Condition C&2(b) that

Let E > 0 be chosen so small that cos(cr,+p) > [/2 for all ( p ( 5 3 ~ , and let n, > no be chosen so large that (8.12) holds and I arg((a2,-1 - z,)/(-Z: - %))I < E for all n > no/2. Then it follows from (8.11) that

for n > n,/2. This proves (8.8) with no := nE and ,u = [/2 for even n. The case where n is odd follows similarly. 0

242 LISA LORENTZEN

PROOF OF THEOREM 2.7. We first observe that since Rl and Vl are bounded, we must have - 1 # V o U V 1 . This means in particular that - 1 # a* Kl U d* VL , and thus also that 0 # d*l.;, Ud*Vl . Let K ( a , , / l ) be a continued fraction from ( E l , E2).

Assume first that a*% = 8 or that d(-h,, d*Vn) does not vanish as n - x. Then the convergence of K ( a , , / l ) follows from Theorem 2.4.

Assume next that a*M/;I # 0 and that d(-h,,, a"V,,) 4 0. Then limsup Ih,/ < x, and thus d(-h,, , dl L',) 4 0 by Condition C* 1 and Lemma 8.1 with Remark R8.1.2. hioreover -h,, E WR for all n > 1 since -1 E W,". (See Lemma 5.2.) It follows therefor? from Lemma 8.2B and D that there exist constants ko > 0 and no E N such that

(8.14) k" 'Qn' < d i a l n s n ( K ) 5 2IQ7, 1 for all n > no.

d(-hill W,) - d(-h,,, aV,,)

Hence d,, := diarn s,, (K) 4 0 if and only if G,, := IQ, ]Id(-h, , , d V n ) 4 0. It follows from Lernma 8.3 that Rr,, := G L n - ~ / G 2 , L is bounded from below by 1 f Xd(-hLn-L, 3l.4) for all n > n o / 2 for some X > 0 and no E W. This means that if ~ ~ = 1 , 1 1 , , RL,, = X, and thus G 2 , 4 0 , then diam s,,(E) 4 0 by (8.14) and the

nestedness (2 .1) of s,,(%). It remains to consider the case where nz=nI , ,2 RL,, < oo.

Similarly, ~f n ~ = , l l , RL,,+ 1 = oo. then G27,+I + 0. and thus, again,

diam S , (V,) --t 0. Assume therefore that both n,"=,lolL Rln and nF==noll R2n+l

converge to finite values. Then C:=,,=,,o d(-h, , dV,,) converges to a finite value. It

follows from Lemma 8.3 and formula (8.14) that 6, := diam S,(K) 4 6 > 0. In fact. since by (8.14) koG,, 5 b,, I 2G,,. we have JQl,J = p,, . d(-h,,, dV,) . where

&,/2 5 p,, 5 6 , l h . Hence C~=r,l,12 l Q 7 , 1 < oo. However, Q,, = h,, (S, , (0) - S,,_ ( 0 ) ) . where {-h21,+J) has all its limit points in

8 4. Hence, {h , , } has no hmit point at 0 , and so x:=,, I S , , ( 0 ) -ST,-1 ( 0 ) / converges

to a finite value. But this means that C:=,, (s,,(o) -S,- ( 0 ) ) converges, and thus, so does {S,, ( 0 ) ) . 0

PROOF OF COROLLARY 2.8. Let K ( a , , / l ) be a continued fraction from ( E l , E l ) with all Ia,,l 5 AI,. and let (q), el) be bounded value sets for K ( a , , / l ) given by

:= V , n B ( 0 , A l ) . where iZI := Ml/y and ,Y := d(-1 , 6 U F). Then C * l and C3,2 hold for these bounded value sets with our M > 0. Hence K(a, , 11) converges by Theorem 2.7. 0

9. Proof of Theorem 2.10 and its corollary

The proof of Theorem 2.10 is similar to the proof of Theorem 2.7. We need a "new Lemma 8.3", though:

LEMMA 9.1. Lrt (q), V , ) be twzn value sets for K ( a , , / l ) , where cx, E al.;]ni3Vl and -1 @ d K , U 3V1 If there e a s t an M I > 0 and a subsequence { n k ) of N such

-

thaf n l ~ := n~ - 7 1 1 . 1 LS bounded and s!,;::, (V,, +,) C C V,,, for all k , where Bk are c~rculnr dzsks wzth radn rk 5 A l l , then


--

PROOF. Clearly (T/;), Vl) are also twin value sets for K(a , , / l ) , and -1 @ V1. By Lemma 5.2 we therefore know that -h,, := S r ~ ' ( c c ) E M7: for all n . In particular this means that h,, # m , and thus S,(z) can be written

It follows from Lemma 8.2C that

Let P k denote the center of BI, and z, , , E dV,,, be such that d(-h,,, ,dV,,,) =

1 h,,, + z,,, 1 . Since

it follows that diam ST,,+, (V, ,,,,) < diarn S,,, (Bk). where S,,, (Bk) is a circular disk with radius

5 diam S,,, (K) . hfl d(-hrLh , aVnA ) + 2hI1'

since x/(2x + A) increases with x when A > 0. This proves the lemma. 0

LEMMA 9.2. Let (V), VL) be t w m value sets uizth 0 6 dl(/;, U dV1. and let rn E N and h 1 > 0. Then there erzsts a d,,, > 0 such that d(S,,,(O). dT/;/;,) > d,,, for every contznued fractzon K(a, , / l ) wzth all la,, I 5 A 1 from the correspondzng elements sets

( E l , E2). A A

PROOF. El := ElnB(O. M) and EL := ELnB(O, AI) are compact sets. Assunle that the infimum of d(S,, (O), d v ) ) with respect to K(a , , / l ) from (El . E L ) is 0. Then there is a continued fraction from (El, E L ) with d(S,,, (O), d v ) ) = 0. In particular sk(dVk) n dVk-1 # 0 for all 1 5 k < m. and thus ak is bounded away from 0. This is impossible since S,,, is an open mapping. 0

PROOF OF THEOREM 2.10. AS in the proof of Lemma 9.1, we find that h,, # x for all n E W. and thus that S, can be represented by (9.2). This also means that S, ( m ) # m for all n. Let m := sup rnk. It follows from Lenma 9.1 that if C d(-h,, , dVnh ) = m , then diam S,,, (V, , , ) 4 0. and thus diam S,, (V, , ) 4 0 by the nestedness of S,,(V,,). In particular S, ( x ) = S,,_, (0) converges.

Assume that d(-h,,, , dVn, ) < m . F'rorn Lemma 8.2B and C it follows that (8.14) holds with ko := 1. Hencr. if {Q,,, Id(-h,,, , dV,,, )) has a subsrqnence comerging to 0, then K(a,, 11) again converges by the arguments above. Assume that no such subsequence exists. that is, /Q,,, 1 = ph d(-h,,, , W,,, ) for some pk > 0.

244 LISA LORENTZEN

where { p k ) is bounded and bounded away from 0. Then C IQ,, / < m. NO^, for j > l

- - Qn, - Qn,

~ l " ~ ) ( m ) + h,, s~Y?(o) + h,, '

where s F h ) ( z ) := z . Clearly, dm in Lemma 9.2 may be taken to be non-increasing with m. Using Lemma 9.2, this shows that

(9.5) ISn,+,(m) - sn, (m)l = IQn, I I I Q n h l IQn, I 5 --

s~?;'(o) + h,, I d ( ( 0 ) V ) d m + ~

for 0 < j < m for some dm+1 > 0 depending only on m, M and (Vo, Vl) . Let first j := m k + l < nz. Then (9.5) implies that C /S,,+,(m) - S,,(m)l < m, and thus {S,, ( m ) ) converges to a finite value f .

Choosing j := jk < m k + l in (9 .5) , shows that also {S,,+,,(m)) converges to f . Hence S , (m) = S,- ( 0 ) converges to f . 17

PROOF OF COROLLARY 2.11. Let K ( a , / l ) be an arbitrarily chosen continued fraction from ( E l , E 2 ) . Then K ( a , / l ) satisfies the conditions in Theorem 2.10 with

( 2 k u ) A

n k := 2kv for all k , since S2" ( 2 ) = S 2 " ( z ) for some continued fraction K ( Z , , / l ) from (El, E2). Hence K ( a , / l ) converges. 0


We shall also here base our arguments on Lemma 8.3. However, our emphasis this time is on the uniformity of (Vo, V l ) . The first lemma is inspired by C6rdova's work [l , Lemma 3.11. Recalling the definition (2.10) of the class R of non-negative functions U J ( ~ ) , it can be stated as follows:

LEMMA 10.1. Let V c @ be open, and let zo E dV \ { m ) , w E R and 6 > 0 , q > 0 be given. Assume that F := dV n B ( z o , q ) is a simple smooth arc with parametric representation (negative orientation w.r.t. V)

where y, ,O E R with 21PI 5 1 /q , and lg(t)l 5 w ( t ) t 2 , lg(t)l < w(t)l t l and R e g ( t ) 2 -1 + 6 for all t E I . Then there exists a continuous, real-valued function p ( z ) for z i n ( C \ V ) n B(zO,q/2) such that the following hold:

A. d ( z , dV) = Im { ( z - zo)ep"} + p ( z ) . [Re { ( z - zo)e-"}]2 for all z E (@ \ V ) n B ( % ! q / 2 ) .

B. To every E > 0 there exists an r > 0 , depending only on E , 6 , q and w , such that + PI < E for all z E (@ \ V ) n B ( z 0 , r ) .

PROOF. Let z E ( C \ V ) n B(zo, z ) be arbitrarily chosen. For simplicity we write

(10.2) X := Re { ( z - z o ) e p q ) , Y := Im { ( z - z ( l ) e - i7 ) , z := d ( z , dV).


A. We define { ( x ) X 2 if X # 0 , p ( z ) :=

r?T//' if X = 0. (Note that 1 - 2 Y P > under our conditions, since IY/ lz - zol < q /2 and 21Pl 5 119.) This proves the expression for x = d ( z , d V ) since ) z - zo/ < q /2 implies that x = lz - z01 = Y if X = 0. To see that p ( z ) is continuous, we only have to check that

x - Y - lim ----- - -P

z - Z , X 2 1 - 2 Y P '

where zl E L n (@ \ 7) and L is the line normal to r at zo, and thus X + 0. for every fixed x > 0 with x < q/2 . Since z E B ( z o , q / 2 ) , whereas r = d V n B ( z g , q ) , it follows that d ( z , d V ) = lz - z ( t l ) l for a z ( t l ) E r. Vi7e have

x 2 = d ( z , d ~ ) ~ = min lz - z ( t ) I2 1

(10.4) = min / X + i Y - t - i p t 2 - g(t)12 t

where u := Reg and v := Img. This minimum is attained for t = t l . That is. 2 ( X - t l - u ( t l ) ) ( - 1 - u ( t 1 ) ) + 2 ( Y - Pt: - v ( t l ) ) ( - 2 P t l - v ( t 1 ) ) = 0 , where J u ( t , ) l < d ( t l ) t ? , lu(t l) l < w( t l ) l t l I , Iv(t1)l 5 w(t1) t . f and b ( t l ) I i w( t l ) l t l I. This means that

since u ( t 1 ) > -1 + 6 , and thus

where the @term is bounded by some function w1 ( X ) . where UJI E R only depends on 6, q and w. Inserted into (10.3) this gives

where also this o-term is bounded by such a function w 2 ( X ) with w2 E R. depending only on 6, q and w. Of course, to let z + zl means to let X + 0 . This changes the value of Y as well. Since x > 0 , we must eventually have Y > 0 , and then

when X is close enough to 0. This proves (10.3).

B. From the arguments above it follows that

246 LISA LORENTZEN

where 21PI < 1/q. IYI < 912. 1x1 < q/2 and lo(X)I < S ( X ) for some S E R which only depends on 6, q and ~ j . Hence

which proves part B, since X I < r and IY / < r when z E B(z0, r ) . 0

The next lemma is crucial for the estimate of d(-h,, dV,,):

LEMMA 10.2. Let hT > 0, and let (I(), Vl) be open twin value sets such that 0 @ d i q ) u dtv1 , K/;, n (-1 - VL) = 0 and Condition C t 2 holds with this M. Further let j E {0,1) and <, E d t VJ n B(0, M ) . Then there exist positive constants r , p and A , depending only on (I(), Vl) and M , such that

(10.5) d(s- '(z), dVJ+l) 2 d(z, 84) -Xd(z , 8 ~ ) ~ for all z E ( - l - T + , ) n ~ ( < ~ , r )

for every a = -C2 E E3+1 with c E F n B(<,,p).

PROOF. We have -1 -TI+, 2 m,. (10.5) holds trivially if z E 8VJ = aWJ, so let z E W,O n (-1 (10.5) is also obvious if d(z,dV,) 2 1 / X for the (large) constant X to be chosen, so let

Let q. x and ~ l ( t ) be the quantities for which Condition Ct2 holds for our h l > 0. Without loss of generality we assume that q > 0 is chosen such that there exists a d > 0 such that d(O,I', ,)> d for all <, E d t ~ , n B(0, M); J = 0 , l . Let further a = -cL E Ej+l with c E VJ n B(<, , p), where p > 0 is some (small) number to be chosen. We require that p < 912. Then the condition that r,, is a simple arc in Ci2(a) implies that d(c, dV,) = Ic - w, 1 for a w, E T, ,. That is, we may write

(10.7) c = w, - iRez7l where y, = y(dV,, w,) and R := d(c,dV,) > 0.

For simplicity we also write

We also observe that if q > 0 and r > 0 are chosen small enough, then c # 0 and 1 ~ 1 2 d/2. Now lz - W , 1 5 iz - <, I + I<, - ci + ic - w, I < r + 2p < q/2 when p and r are chosen small enough. That is, z E H73" n B(w,, q/2), and since Ct2(a) holds, it is a consequence of Lemma 10.1 (with V := V, and zo := w,) that

(10.9) x := d(z, dV,) = Y + (-P, + v)X2 and thus Y = x + (PJ - v)X2,

where 0, := - ir;(dVJ, w,) and v E IR can be made arbitrarily close to 0. independently of <, E n B(0, i l l) and w, E I', , , by having z close enough to w, ; that is, by choosing p and r small enough. Moreover. s p l ( z ) E sp l ( -1 - F + l ) & -1 -T C m,+L and


where / z - wjI < r + 2p. When p > 0 and r > 0 are small enough, we thus have 1s- l ( z )+l+w, / < 912. B y using Lemma 10.1 with V := -1- W,", zo := -1-u!,, and the representation z = -1 - q ( - t , w,) for a(-1 - W;) we therefore get

where a E IR is close to 0 if s-' ( z ) is close to - 1 - w, . Straightforward computation. using (10.7) and (10.8), shows that

(w, - iRez?j )" ( s - l ( z ) + 1 + w, )epZ?~ = + w,) e- ' .~

w, + ( X + i Y ) e " ~

= X + i { ~ + 2 R + ((Y + R)' - x 2 ) q 2 - 2 X ( Y + R ) q l }

where we have used the notation

e 2 Y ~ ~ ' Y J vi :=Re---, qz :=Im--- and T : = I X + i Y I + R I r + 3 p ,

W 3 W~

and where IRe m(T) I 5 M2T2 and IIm m(T) I < M3T"or some constants Mk > 0 independent of <3 and w, for k = 2,3. Inserted into (10.10), using the expression (10.9) for Y , this gives

where 10(?')) < G F v o r some %? > 0 with T := /XI + z + R 5 r + 4p. From Condition Ct2(b) we know that 2p, - 11.2 > x > 0. Let r > 0 and p > 0 be chosen so small that we also have Iul 5 x / 8 , la1 I x / 8 and %?T 5 x / 4 when

248 LISA LORENTZEN

z E ( -1 - F+l) f' B ( C 3 , r ) and c E Fn B ( & , p ) . Then (10.11) implies that

> x + 2 R - C ( x + R)' where C := -ql + 1 ~ 1 1 + - - T2 I ' 8

Hence, if we make sure that also 3pC < 2 and set X := l l p . then C 5 X and 2 R - C ( 2 x R + R2) 2 R ( 2 - C ( 2 p + p) ) > 0 for all R < p and x < 1 /X , and the result follows. 0

PROOF O F THEOREM 2.12. We first observe that since -1 $! VO U V 1 , and thus in particular -1 6 d*& U d*Vl . we also have 0 $! d*q / ; , U d*Vl . Let K(a, , 11) be a continued fraction from (El, E2) with all la,, I 5 M for some M > 0 , and let M 1 := M / d ( - 1, 6 U F). It follows from Proposition 2.2 that K ( a n / l ) also has

A A A

the bounded value sets (h, Vl ) given by V, := V,, n B ( 0 , M I ) . Since 0 E GI, we know by Lemma 5.2 that h,, E 1 + R,+l for all n. and thus Ih,l < 1 + M 1 for all n . By Theorem 2.4 it follows that d i a m ~ , , ( C , ) + 0 if d(-h,, d*C,) f t 0. Since

A

S,, (V,,) C S , , 1 (V,,-I), this proves the result for this case. Let d(-h,. d * c , ) + 0. It is a consequence of Lemma 8.1 and Remark R8.1.2

that

(10.12) d(-h, , , dTV,,) + 0 , ( 1 - h,, - h,-1) + 0 and (a,, + h t - , ) + 0 ,

and thus d(-h,, d t V n ) + 0 and d(a,,, - ( d i R , - l ) 2 ) -t 0. Let X > 0 , 7- > 0 and p > 0 be some constants for which (10.5) holds. Let further c,, be the square root of -a, closest to -h,-l, and no E N be so large that d(c,,, dtV, ,- l) < min{p,p, 7-12} and Ic, +hn- 1 / < 7-12 for all n > no. Then c, E V,,- 1 for n > no with d(c,, d t V , , 1 ) < p. Let <,-l E dtVn-l besuch that d ( c , , d t ~ , - ~ ) = /c,-<,-11. Then Ih,-l+<,-lI < r ; i.e., -h,-1 E ( -1 - V,) n B(<,-,, r ) . It follows therefore from (10.5) with z := -h,-l that

dl, > d,-l - ~ d : ~ - , where dk := d ( - h k , d V k ) .

For sufficiently large no we have d,, < 1 / ( 2 X ) for all n > no, which shows that

1 1 1 X I-+-

1 (10.13) - 5 5 - +2X for n > no.

dn+, dl, - Ad?, d , 1 - Ad, d,,


Repeated application of (10.13) shows that

and so C d, = C d(-h,, dV,) = m. Since Ct2 implies C*2, it follows therefore from Lemma 8.3 that G,, := IQ,l/d(-h,, dVn) 4 0. Since diam S,,(Vn) 5 2G, by Lemma 8.2B, the result follows. 0

PROOF OF COROLLARY 2.13. It follows from Theorem 3.1 that E, = -C: where C, C (1 +W,) n (-1 - Wj) for j = 1,2. Since -1 - WTj = V,-, in our case, we thus have Cj C: 7 for j = 1,2, and the result follows from Theorem 2.12. 0

PROOF OF COROLLARY 2.14. Since m E 8x1, 0 E V/;, and -1 @ 6, and dVo is a Jordan curve, the expression (2.14) for E, follows from Theorem 3.3. Hence E," f 8 , and the result follows from Corollary 2.13.

11. Lange's strip convergence regions

The notion strzp regzon was introduced by Lange 1111 to describe a simply connected, closed domain C in @ whose complement @ \ C consists of exactly two disjoint components. A typical example is the strip region C, in Thron's Uniform Parabola Theorem, as given in (1.15). This strip region C,, which lies on and between two parallel1 lines, was actually the inspiration for some strip regions suggested by Thron in 1944 [26]. (Thron's strip regions will not be considered in this paper, because he had -1 E dV and 0 E dV.) Likewise, they inspired Lange's strip regions [ll]. We recall that P, := -C: is the local convergence set of Thron's

- Uniform Parabola Theorem. Moreover, C, = V, n (-Pa) \ {m) . Hence, the two components DL and D z of @ \ C, satisfy DL = @ \ V, and D: = V, \ C,. We find the same pattern in Thron's and Lange's strip regions. In this section we shall look closer at some properties of Lange's strip regions.

His first family of strip regions is his class of Worpitzky Strips; i.e., strip regions C containing the Worpitzky Disk J z J < 112. The boundaries of these strips are made up from rays and circular arcs. If we change Lange's notation to match ours. these strip regions can be described as follows:

THEOREM 11.1. (The Worpitzky Strips [ll, Thm. 3.1, p. 2241). Let 0 5 a < 7r/3 be a given constant, and let V(a ) be the Jordan domain in @ with 0 E V(a) an,d boundary made up of the following rays and circular arcs:

L a ) : z = - l + e m ( ; + + t ) f o r t 2 0 ,

( a ) : z = - l + $ e z a t for O I t s l , (11.1) r(I(a) : z = --le-znt

2 for - 1 I t L 0 ,

Lo(&) : z = e m ( - ; + i t ) f o r t s o .

Then V(a) is a value set for contznued fractions K ( a n / l ) . and V(a ) has corresponding element set E := -C2, where C := V(a) n (-V(a)) \ { m ) . Moreover, the complex conjugate of E is the element set correspondzng to the complex conjugate

of V(a ) .

250 LISA LORENTZEN

1. Ll (a) and ro(a) are parts of the two circles with radius and centers at -1 and 0 respectively. They are connected at z = - i. The ray L, (a) is tangent to the circle with center at j , and it takes off where the arc I?, ends, at an angle cw with the imaginary axis. Clearly dV(cw) = d*V(a) = d iV(a ) is a Jordan curve in e. It has continuous curvature, except at the points where the parts of dV(cu) are "glued together", where dV(cu) only has one-sided curvature. A picture of the set C can be found in [ll].

2. For a = 0 the Worpitzky Strip coincides with the strip Co in Thron's Uniform Parabola Theorem with a = 0. (See (1.15).)

3. We have 0 E V, -1 @ V , oo E dV, V = @ \ (-1 -V) and dV = -1 - dV for V := V(cu). Hence Theorem 11.1 is consistent with Theorem 3.3.

Lange's second class of strip regions is his class of Trancendental Strips. Again we have changed Lange's notation to match ours.

THEOREM 11.2. (The Trancendental Strips [ll, Thm. 4.1, p. 2291). Let -1 2 < - d 5 4 be a given constant, and let Vd be the Jordan domain in @ with 0 E Vd and boundary

1 2d (11.2) dVd : z = zd(t) := -- - - tan- ' t + it for - x < t < ca.

2 7r

Then Vd is a value set for continued fractions K(a, / l ) with corresponding element set E = -C2, where C := % n (-vd) \ {m).

REMARKS R11.2.

1. Clearly, z E dVd with d # 0 iff z = -; + z + iy with y = - t a n ( ~ z l 2 d ) . It is also clear that K d is the complex conjugate of Vd. For d = 0 the Trancendental Strip coincides with the strip Co in Thron's Uniform Parabola Theorem. As a consequence of Theorem 4.4 we find that Vd is the unique open value set for E in Theorem 11.2.

2. In his paper [26] Thron defined conditional convergence regions E (or C ) for continued fractions K(a, 11) to be regions such that every continued fraction K ( a , / l ) from E has the property that K(a,,/l) converges if and only if

(We recognize (11.3) from the Stern-Stoltz Theorem as stated in (1.17).) By appealing to the theory of normal families, he proved that a subset C, of his strip region C , where certain parts of the boundary of CE stay a distance E > 0 away from dC, gives a conditional convergence region for continued fractions K(a, / l ) . Similar results were proved by Lange [ll] for his Worpitzky Strips and his Trancendental Strips. Lange also pointed out that C6rdova's Conver- gence Theorem applies to the Trancendental Strips.

We shall prove that Lange's strip regions are convergence sets locally for continued fractions K(a,,,/l). That is, we allow a, = -c:,, where c,, is on or close to the boundary of the strip, at the expence of having {a,,) bounded. Lange's two fanlilies of strip regions require slightly different techniques, since the Worpitzky


Strips do not have continous curvature at the three points where the arcs are "glued together". Moreover, C6rdova's conditions Y 2 and Y 3 do not hold either for this value set. It seems reasonable to expect that Corollary 2.14 can be made valid for cases where we only have one-sided curvature; that is, for cases where Condition C t 2 is slightly weakened. Since V ( a ) satisfies the weaker C6rdova Condition C * l , this would have made it possible to apply Corollary 2.14 to the Worpitzky Strips. However, at the present point, we rather apply Corollary 2.9:

THEOREM 11.3. Let cu E R with 0 < a < 7r/3, and let V = V ( a ) and E be as i n Theorem 11 . I . Then the following holds.

A. E is a best convergence set locally i n the sense that E is a convergence set locally, but E U { a ) is not a convergence set locally for any a @ E U {m).

B. V is the unique open value set for E , and V is the best limit set corresponding to E . Actually, every w E V \ ( 0 , m) is the value of a convergent continued fraction K ( a , / l ) with a1 E dE1 and azlL = - ( 1 + 02, a2",+1 = -(2 for n > 1, where < E d V .

C. Parts A and B also hold i f we replace E and V by their complex conjugates.

PROOF. In view of Corollary 2.9 it suffices to prove that V = V ( a ) satisfies the conditions (ii) - ( i i i ) in this result, to conclude that E is a convergence set locally. The case cu = 0 is covered by Thron's Uniform Parabola Theorem.

We shall first prove that the C6rdova Condition C* 1 is satisfied for v) = V1 = V . Evidently both d V ( a ) and d( -1 - V ( a ) ) are starshaped with respect to the origin; i.e., dV(cu) is starshaped with respect to both 0 and -1. Hence it follows from Proposition 3.4 and Remark R3.1.4 that it suffices to prove that

where Co(t) is some representation of d V ( a ) with negative orientation with respect to V ( a ) . We choose to use (11.1) piecewise. Straightforward computation shows that (11.4) holds on each part L,(cr), I',(cr); i = -1,O. This implies that (11.4) holds for the total boundary.

( i i i ) ( a ) holds trivially. To check ( i i i ) ( b ) we just have to prove that the line z = C + < ( l + <) t ; t E R, is not a tangent to d V at ( for any i E d V , since y ( d V , i) varies continously with C. This is again a consequence of (11.4). Hence, E is a convergence set locally by Corollary 2.9. The optimality in A and B follows from Theorem 4.1, Theorem 4.3 and Theorem 4.4. Part C follows trivially. 0

It is easy to establish twin versions of Lange's Worpitzky Strips. We have for instance:

THEOREM 11.4. Let a , r E R be given constants with 0 < a < 7r/2 and 0 < r < 1 such that max{r, 1 - r ) < cosa, and let &(a , I-) be the Jordan domain with

252 LISA LORENTZEN

0 E &(a, r ) and boundary given by

( -1 + e z a [ l - r + i ( t - I ) ] for t 2 1, for 0 < t 5 1,

for - 1 5 t < 0 ,

Further, let V l (a , r ) := @ \ ( -1 - V o ( a , r ) ) . (That is, Vi(a, r ) = VO(Q, 1 - r ) . ) Then the following hold.

A . ( h ( a , r ) , V l ( a , r ) ) are twin value sets with corresponding element sets -

( - C f , 4';) given by Cj := V 3 - 1 ( ~ , r ) n ( - V j _ l ( a , r ) ) \ { c o ) for j = 1 , 2 .

B. ( - C f , - C z ) are best twin convergence sets locally for continued fractions K ( a , / l ) i n the sense that ( - C f , -c;) are twin convergence sets locally, but (-Cf U { a ) , -C;) are not twin convergence sets locally for any a $! -C? U

{ w > . C. ( V o ( a , r ) , Vl(a, r ) ) are the unique open twin value sets for ( - C f , -C;)? and

their closures (Vo(a, r ) , V 1 ( a , r ) ) are the best twin limit sets corresponding to ( - C f , 4';). Actually. every w E V o ( a , r ) \ ( 0 , ~ ) is the value of a convergent continued fraction K(-c: /1) with cl E d x , and c2, = -1 - C , c2,+l = C f o r n > 1, where C E d V o ( a , r ) .

D. Parts A , B and C hold i f we replace K/;,(a, r ) (and thus Vl (a . r ) ) by its complex conjugate.

The proof is similar t,o the proof of Theorem 11.3, and based on trivial, but lengthy computation which is not repeated here. The choice r := makes Vo(a, i) = Vl (a , $) which is equal to V ( a ) in Theorem 11.3. The possibitlity of varying r , means that we may vary the radii of the two circular arcs L l ( a ) and ro(a) in (11.1).

As pointed out in [ll], C6rdova's Convergence Theorem applies to Lange's Trancendental Strips. We can also use Corollary 2.14. Either way, we get:

THEOREM 11.5. Let -; < d 5 $, and let V = Vd and E be as i n Theorem 11.2. Then the following holds.

A . E is a best convergence set locally i n the sense that E is a convergence set locally, but E U { a ) is not a convergence set locally for any a @ E U {m) .

B. V is the unique open value set for E , and V is the best limit set corresponding to E . Actually, every w E I / \ ( 0 , co) is the value of a convergent continued fraction K ( a , / l ) with a l E d E , and an,, = - (1 + 02, a.27L+1 = -C2 for n 2 1 , where < E d V .

C. 1/ is a uniform value set for every continued fraction K ( a , / l ) from E with bounded element sequence.

PROOF. Let K ( a , / l ) be a continued fraction from E with a bounded element sequence {a,}. We shall first prove that K ( a , / l ) converges. This follows from Thron's Uniform Parabola Theorem if d = 0. Since V-d is the complex conjugate of ri;,, it suffices to consider the case where d > 0. For that case. Lange [ll] has proved that Vd is starshaped with respect to the origin. Since dVd = - 1 - dVd, it is also starshaped with respect to -1. Lange also proved that 8Vd is logarithmically


convex. Hence it follows from Proposition 3.4 that the C6rdova Condition Y2 is sat,isfied. Y1 holds trivially. It remains to prove that Y3 is satisfied.

According to Remark R5.2.2 it suffices to prove that

where ?3 := y(dV, C3). That is,

which holds if and only if

where zd is given by (11.2). Since 1 + zd(t) = -zd(-t), it suffices to prove the first inequality in (1 1.7). We have

Straightforward computation shows that f (t) := tan-' t - t / ( l + t2) is increasing for t E R. For t > 0 this shows that

where F(0) = 0 and F(co) = $. Straightforward computation shows that F ( t ) has just one critical point. t = I/&. and that ~ ( 1 1 4 ) = -0.1. Hence, Hd(t) > 1 - a . 0 . 1 > 0 fo r t > 0. 2 T

For t < 0 we have

Hence it follows from C6rdova's Convergence Theorem that E is a convergence set locally. Also Part C follows from C6rdova's result. The optimalities in parts A and B follow from Theorems 4.1, 4.3 and 4.5. 0

Lange's Trancendental Strips can also be extended to twin strips. The simplest extension is possibly the following, where we have just introduced a second parameter r:

254 LISA LORENTZEN

THEOREM 11.6. Let d , r E R be given constants with Id1 < min{r, 1 - r ) , and let Vd.r be the Jordan domain with 0 E Vd,r and boundary given by

(11.9) 2d

z = ~ d . ~ ( t ) := -r - - tan-' t + i t ; t E R. ll

Then the followzng hold.

A . (Vd r , Vd are - twzn value sets wzth correspondzng element sets (-Cf, -Ci) gzven by Cj := Vd,r, n (-Vd r,) \ {co) for J = 1,2, where rl := r and 7-1 := 1 - r .

B. (-Ct ,-Ci) are best twin convergence sets locally i n the sense that (-C;, -Ci) are twin convergence sets locally, but (-Cf U { a } , -C;) are not twin convergence sets locally for any a $2 -Cf U {co).

C. (Vd.r, V ~ . I - ~ ) are the unique open twin value sets for (-Cf, -C:), and (Vd.r, Vd,l-r) are the best twin limit sets corresponding to ( -Cf , -C:). Ac- tually, every w E pd.,\{O, co) is the value of a convergent continued fraction ~ ( - c ; / l ) with cl E aVd,,, and c2, = -1 - <, cn,+l = C for n > 1, where C E 8Vd.r.

D. (Vd,r, Vd,l-r) are uniform twin value sets for everg continued fraction K(a,/ l ) from (-Cf, -C;) with a bounded element sequence.

PROOF. It suffices to prove the theorem for the case where 0 < d < min{r, 1 - r ) . Further, it suffices to prove that Irn(id,,Zd,,) < 0 and that Hd,,(t), defined in the same manner as Hd(t) in (1 1.8), is positive for every fixed 0 < r < 1. This follows from arguments similar to the ones in the proof of Theorem 11.5, since

References

A. Y. Cbrdova, A convergence theorem for continued fractions, Thesis. Bayerischen Julius - Maximilians - Universitat Wiirzburg. (1992). J . D. De Pree and W. J . Thron, On sequences of Moebzus transformatzons, Math. Zeitschr. 80 (1962), 184-193. L. Jacobsen, Some perzodzc sequen,ces of circular convergence regzons, In: Analytic Theory of Continued Fractions (eds.: W.B. Jones, W. J.Thron and H.Waadeland), Lecture Notes in Math., Springer Verlag 932 (l982), 87- 98. L. Jacobsen, General convergence of continued fractions, Trans. Amer. Math. Soc. 294(2) (1986), 477-485. L. Jacobsen, A theorem on szmple convergence regzons for contznued fractzons K(a , , / l ) , In: Ananytic Theory of Continued Fractions 11, (ed.: W.J.Thron), Lecture Notes in Math., Springer Verlag 1199 (1986), 59 6 6 . W. B. Jones and W. J . Thron, Twzn-convergence regions for corztinued fractions K(a , , / l ) , Trans. Amer. Math. Soc. 150 (1970), 93-119. W. B. Jones and W . J . Thron, Continued fractzons. Analytzc theory and applzcatzons., En- cyclopedia of Mathcmatics and Its Applications. No 11, Addison - Wesley (1980). W . B. Jones, W . J. Thron and H. Waadeland, Truncation error bounds for contznued fractions K(a, , / l) wzth parabolzc element regzons, SIAM J . Numer. Anal. 20 (1983), 1219-~1230. L. J . Lange, On a family of twzn convergence regions for continued fractions, Illinois J . Math. 10 (1966), 97~-108. L. J. Lange, A uniform twzn parabola convergence theorem for contanued fractzons, J . Math.

t

Ananl. and Appl. 188 (1994), 985-998. L. J . Lange, Strzp convergence regions for continued fractions, In: Continued Fractions and Orthogonal Functions (eds.: S. Clement Cooper and W. 3. Thron), Lecture Notes in Pure and Applied Mathematics, Marcel Dekker (1994), 211 - 231.

I


12. L. J . Lange and W . J. Thron. A two parameter family of best twzn convergence regions for continued fractions, Math. Zeitschr. 73 (196O), 295-311.

13. W . Leighton and W. J. Thron, O n value regions of contznued fractzons, Bull. Amer. Math. Soc. 48 (1942), 917 920.

14. W . Leighton and W. J . Thron, Continued fractions wzth complex elements, Duke Math. J . 9 (1942), 763 -772.

15. L. Lorentzen, Compositions of contractions, J . Comput. Appl. Math. 32 (1990); 169 178. 16. L. Lorentzen, Bestness of the parabola theorem for contznued fractions, J . Coniput. Appl.

Math. 40 (1992), 297-304. 17. L. Lorentzen, Propertzes of lzmzt sets and com~ergence of contznued fractions, J . hlath. Anal.

and Appl. 185(2) (1994), 229 255. 18. L. Lorentzen, Convergence of composztions of self-mappings, Ann. Univ. Marie Curie Sklo-

dowska (to appear) 19. L. Lorentzen, Convergence of limzt periodzc contznued fractzons K(a, , / l ) of parabolzc type,

(in preparation). 20. L. Lorentzen and St. Ruscheweyh, Simple convergence sets for contznued fractzon,s K(a , , / l ) ,

Math. Anal. and Appl., 179(2) (1993), 349 - 370. 21. L. Lorentzen and H. Waadeland, Continued fractions with applicatzons, Studies in Compu-

tational Mathematics 3 (Elsevier Science, 1992). 22. J . F. Paydon and H. S. Wall, The contznued fraction as a sequence of h e a r fractzonal

transformations, Duke Math. J . 9 (1942), 360-372. 23. G. Piranian and W . J . Thron, Convergence propertzes of sequences of h e a r fractzonal trans-

formations, Michigan Math. J . 4 (1957), 129-135. 24. W . T . Scott and H. S. Wall, Value regions for contznued fractions, Proc. Amer. Math. Soc.

47 (1941), 580-585. 25. W . J. Thron, Two famzlzes of twin convergence regions for contznued fractzons, Duke Math.

J . 10 (1943), 677-685. 26. W . J. Thron, A family of simple convergence regzons for continued fractions, Duke Math. J .

11 (1944), 779-791. 27. W . J. Thron, Zwzllingskonvergenzgebtete fur Kettenbruche 1 + K ( a , , / l ) , dwen eines dze

Krezsscheibe iazrL-l 1 < p2 ist, Math. Zeitschr. 70 (1959), 310-344. 28. W . J. Thron, O n parabolic convergence regions for contznued fractzons, Math. Zeitschr. 69

(1958), 173 - 182. 29. H. Waadeland, Boundary versions of Worpitzky's theorem and of parabola theorems, In:

Analytic Theory of Continued Fractions I11 (cd.: L. Jacobsen), Lecture Notes in hlath. , Springer-Verlag 1406 (1989), 135 142.

DIVISION O F ~ I A T H E M A . T I ( . A I . S C I E N ~ E S , NOR\VE(;IAN I 'NIVERSITY O F S < . I E N ( ' E AN11 . I ' I< ( 'H- NOLO<:Y. N 7 4 9 1 TRONUHF:IM. NORWAY

E-mail address: lisaomath .ntnu.no

Contemporary Mathematics Volume 236. 1999

Strong Stieltjes Moment Problems

Olav Njgstad

Dedzcated t o Professor L.J. Lunge o n the occasion of hzs 70th birthday.

ABSTRACT. For a given strong Hamburger moment problem the quasi- orthogonal Laurent polynomials determine (for each z outside the real axis) a system {A,(z)) of nested disks. The moment problem is determinate exactly in the limit point case, i.e., when the intersection of the disks reduces to a single point. When the corresponding strong Stieltjes moment problem is solvable, the pseudo-orthogonal Laurent polynomials determine another system {D, (2)) of nested disks. In this paper we discuss properties of the system {Dn(z)) and its relationship to the system {A,(z)). In particular we study applications to the question of determinacy or indeterminacy of the strong Stieltjes moment problem.

1. Preliminaries

The Stieltjes transform F, of a finite measure y on (-m, oo) is the function

The moments c, of a positive measure y are defined as

(1.2) c,, = 1 tnc ip ( t ) , n = 0. *I, * 2 . . . .

-z

if these integrals exist. The strong Hamburger moment problem ( S H M P ) for a given sequence {c, : z = 0, il, 1 2 , . . . } of real numbers is to construct and study measures y satisfying (1.2). The strong Stzeltjes moment problem (SSLIP) is to construct and study measures with support in [0, m) which satisfy (1.2). A moment

1991 Mathematzcs Subject Classification. AMS Subject Classification: 30E05, 42C05, 44A60. Key words: Strong moment problems, Orthogonal Laurent polynomials.

@ 1999 American Mathematical Socicty

257

258 OLAV NJASTAD

problem is determinate if there exists exactly one solution, indetermznate if there exists more than one solution.

An important tool in the study of strong moment problems is the theory of orthogonal Laurent polynomials. A Laurent polynomial is a finite linear combination of monomials z'" n = 0, 5 1 , &2, . . . . We define

(1.3) A,., = Span{zP, . . . , zq)

where p and q are integers, p < q. Thus the space A of all Laurent polynomials is the union of all the spaces A,,,.

Let the sequence {c,,) be given. For simplicity of notation we shall assume the sequence normalized such that co = 1. We define a linear functional M on A by

When the SHMP is solvable, this linear functional gives rise to an inner product (,) on the space of real Laurent polynomials by the formula

(1.5) ( f td = MIS .9l.

Then for all solutions ,u of the SHMP we have 3i

(1.6) (f, s) = 1 f ( t ) d f ) d , u ( t ) . 3 C .

The focus of this paper is on the study of solutions of the SSMP. W e shall in all that follows assume that the SSMP zs solvable. Many of the basic results in the theory are, however, valid as long as the SHMP is solvable.

For the basic theory of orthogonal Laurent polynomials and strong moment problems we refer to [5]-[6]. [lo], [12], [14]-[18], [23]-[29]. For the analogous theory of orthogonal polynomials and classical moment problems, we refer to e.g. [I]-(41, [7]-[9], [13], [19]-[22], [30]-[35], [37].

2. Orthogonal Laurent polynomials

In this section we sketch some fundamental properties of orthogonal Laurent polynomials and solutions of the SSMP. For details, see the references above.

By orthonornlalization of the sequence

(2.1) (1, z l , Z, 2 -2 , 22,. . ' , Z-7n,z7n,. . . )

with respect to the inner product (1.5) we obtain a sequence {cp, : n = 0,1 ,2 , . . . ) of orthonormal Laurent polynomials. They satisfy

and may be written in the form

STRONG STIELTJES MOMENT PROBLEMS

(with (PO = uo = 1). The coefficients v, are called leadzng coeficzents and the coefficients u,, trazlzng coeficzents. They satisfy the inequalities

(2.6) 212111 > 0 , u2177+1 > 0

(2.7) ~ 2 , n > 0 u27n+l < 0. (See e.g. [24, p. 3231). The associated orthogonal Laurent polynomials $,, are defined by

(the functional operating on its argument as a function of t ) . We note that

(2.9) $0 -- 0, $2nj E A-m rn-1, $ L ~ + I E A-(m+l) ,n l -~

The sequences of functions {cp,), {y,,) satisfy the following three-term recurrence relation:

(See e.g. [16. p. 3651, and [5, p. 191, [6, p. 371 where different. but equivalent formulations of the recurrence relation are given in the more general setting of a regular SHILIP. Cf. also [24, p. 3221. (In [24, p. 3221 initial conditions are given for the general situation. These conditions reduce to (2.12) when co = 1. which we have assumed in this paper.) The coefficients f,,, g,,, h,, are given in terms of the leading and trailing coefficients by

260 OLAV NJASTAD

(cf. e.g. [24, p. 3221). It follows from (2.6)-(2.7) that

The zeros <;I, k = 1,. . . n , of p, are simple and positive. A discrete measure

p (n ) with positive masses xP) at the points <PI, k = 1,. . . , n, is defined through a Gaussian quadrature formula. These measures satisfy

M [ L ] = ~ ( t ) d p ( ~ ~ ) ( t ) for L E A-2m.2rrL-1 S 0

(2.20) MIL] = / ~ ( t ) d p ( ~ ~ + ~ ) ( t ) for L E A-(2,,+l).zni.

0

See e.g. [15, p. 5411. The sequences {p(LnL)) and {p(""'tl)} converge to measures p(") and p(") which are solutions of the SShIP. These solutions p(") and p(X) are called natural solutzons of the SShIP. The SSMP is determinate if and only if p(') and p(X) coincide, i.e., if and only if the whole sequence {p(7L)} converges. See [6, p. 49 - 501, [18. p. 5161, [24, p. 3381.

The approxzmants of the continued fraction defined by the recurrence formulas (2.10)-(2.12) are the quotients y,/p,,. The sequences { - s} and { -

,,.+I(., } converge in the complex plane outside the non-negative real axis to the Stieltje transforms F,ci,, (z) and FA,(,, (2) (which coincide if and only if the SSMP is determinate). See [18, p. 5151, [24, p. 3381.

3. Quasi-natural and pseudo-natural solutions

For more information on the topics treated in this section we refer especially to [23]-[27], [29].

The quasi-orthogonal Laurent polynomials 9, (z, T ) of order n are defined by

Here T is a parameter, r E R = R U {M). (For r = X, p,,(z, T) means -(+I)" p,,- 1 ( z ) . Similar conventions are used throughout. Also p,, ( 2 , - W) =

p,(z, m), etc.) The assoczated quasz-orthogonal Laurent polynomzals &(z, T) of order n are defined by

STRONG STIELTJES MOMENT PROBLEMS

T (3.4) 7J~21,,+, ( 2 , ~ ) = V h + L ( 2 ) - ; v2rn ( 2 ) .

The quas~-approzzmants of the continued fraction determined by forinulas (2.10)- (2.12) are the quotients

The zeros < r ) (~ ) . k = 1,. . . , n, of p, ,(z , T ) are real and simple. and at least ( n - 1 of them are positive. See [24. p. 3261. A discrete measure p;) with positive

masses x ~ ' ) ( T ) at the points < l 1 ' ) ( r ) , k = 1, . . . , n, is defined through a certain quadrature formnula. Thcse measures satisfy

x

(3 .7) n r [ q = 1 ~ ( t ) d ~ y ~ ~ ~ + l ) ( t ) for L t A -,.,, ,,,,.

- X

See [ I s , p. 548-5491. [24, p. 3271. The quasi-approximant R, , (z ,T) is the negative of the Stieltjes transform of the

measure p:'). For each z in the open upper half-plane UJ, the AIoebius transformation

maps the extented real line @ onto a circle in U. We use the notation A , , ( z ) for the open disk bourldecl by this circle, d A , , ( z ) for the circle itself, and a,, ( z ) for the closed disk A,, ( 2 ) U 3A,, ( z ) . Thc points in ~ A , , ( z ) are exactly the values at z of

( the Stieltjes transforms of the measures p:'). The sequence {A,, (z)) is nested, i.e., A,,+l ( z ) C A,, ( z ) . \Ye define

X

(3 .9) a x ( z , = n &,(.I. 72=1

Then a , ( 2 ) consists of a single point for each z (the l z m ~ f pomt case) or of a proper closed disk for each z (the lzmzt czrcle case). The points in a , ( z ) arr exactly the values at z of the Stieltjes transforms of all solutioris of the SHMP, i.e.,

(3.10) a , ( z ) = { Z L , = F, , ( z ) : p is a solution of the S H M P } .

See [24, p. 3271. [29. p. 181. The points on the boundary dA,(z ) are exactly the values a t z of all Stieltjes

transforms of solutions p obtained as limits of subsequences of sequences { p g ; ) ) . Such solutions are called quasz-natural solutzons. Thus

(3.11) a A , ( z ) = {w = E;,(s) : p is a quasi-nat,ural solution of the S H M P )

262 OLAV NJASTAD

All the zeros of , , ( z , r ) are positive if and only if r E [h27rr, no] when n = 2 m and if and only if r E [-no, hLr, ,+l] when n = 2 m + 1. See [24, p. 3261. The

( corresponding measures ,LL$) satisfy

x

(3.12) A l [ L ] - 1 L ( t ) d p p n ' j ( t ) for L E A-2in2,r12

0

(3.13) A q L ] = 1 ~ ( t ) d p ~ " + l ) ( t ) for 1, E A - 2 n i . ~ ~ ~ -

0

The values at z of the Stieltjes transforms of these measures cover a subarc R,,(z) of ah, , ( z ) . These arcs tend to a limiting subarc R,(z) (which may reduce to a single point) of a A x ( z ) as n tends to infinity. This arc consists of the values at z of the Stieltjes transforms F, of quasi-natural solutions p of the SSMP. (That is: quasi-natural solutions of the SHMP which are also solutions of the SSMP.) Thus

(3.14) R,(z) = {w = E;,(z) : / I is a quasi-natural solution of the SSMP).

See [24, p. 3301. The end points of R , ( z ) are the values F,(o, ( z ) and Fp(,, ( 2 ) .

The pseudo-orthogonal Laurmt polynomzals an ( z , r ) of order n are defined by

and the associated pseudo-orth,ogonml Laurent polynomials Q,,(z, r ) are defined by

The pseudo-approximants of the continued fraction determined by the formulas (2.10)-(2.12) are the quotients

The zeros <il')(r) of a,, ( z , r ) are real and simple, and at least n - 1 of them

are positive. See [24, p. 3311. A discrete (no t necessarzly posztzve) measure v?)

with masses K:') ( 7 ) at the points <in) ( T ) , k = 1, . . . , n , is defined through a certain quadrature formul~. These measures satisfy

(3.18) h l [ L ] = ~ ( t ) d v ? " ) ( t ) for L E A - ( L m - ~ ) ~ ~ n - 1 i -x

x

(3.19) M [ L ] = J ~ ( t ) d v ; ~ " ' + ' ) ( t ) for L E A-(2,,-lj,2n.+~

x

STRONG STIELTJES MOMENT PROBLEMS 263

See [24, p. 3311. The pseudo-approximant S, ( z , r j is the negative of the Stieltjes transform of the

measure u p ) . For each z in U, the Moebius transformation

maps onto a circle dD,(z) in U. We denote by D n ( z ) the open disk bounded by this circle and by D,, ( z ) the closed disk D, ( z ) u d D , ( 2 ) . The points in dD,, ( 2 ) are

exactly the values at z of the Stieltjes transforms of the measures up). All the zeros of a,, ( z , T ) are positive if and only if T E [-co, gznL] when n = 2 m

and if and only if r E [g2,,,+l, m] when n = 2 m + 1. See [24, p. 3301. In these

cases, and only zn these all the weights K ~ ) ( T ) , k = 1 , . . . , n, are positive. The

corresponding positive measures u p ' satisfy

(3.21) M [ L ] = ~ ( t ) d v y " ~ ( t ) for L E A- (2m-~) .2m- i 7 0

(3.22) M [ L ] = ~ ( t ) d u ! 2 " + ' ) ( t ) for L E A-(2m-1).2,n+l. i o

The values at z of the Stieltjes transforms of these positive measures cover a subarc r , ( z ) of dD,,(z) . Furthermore, the arc r , ( z ) consists of exactly that part of d D n ( z ) which is contained in An-l ( 2 ) . Similarly the arc R,(z) consists of exactly that part of d A , ( z ) which is contained in D,-I ( 2 ) . I.e.,

The sequence {D,, ( 2 ) ) is nested. We define

This set is a single point or a proper closed disk. It can be shown that an invariance result holds also for these sets: Either D,(z) is a single point for every z E UJ or D , ( z ) is a proper disk for every z E UJ. See [28]. The arcs r , ( z ) tend to a limiting subarc I?, ( z ) (which may reduce to a single point) of d D , ( z ) as n tends to infinity. The points of r , (z ) are exactly the values at z of the Stieltjes transforms of all

solutions p of the SSMP obtained as limits of subsequences of sequences {u!:)). See [24, p. 3401. Such solutions are called pseudo-natural solutzons. Thus:

(3.26) r , ( z ) = {w = F,(z) : I*, is a pseudo-natural solution of the SSMP).

264 OLAV NJASTAD

(Recall that we rlo not obtain solutiorls of the SHhlP from sequences {v!::)} where T, does not eventually belong to the T-intervals [-m, g ~ ~ , ] and [gLrn+l, m].) The end points of r, (z) are the values F,,l~i ( 2 ) and I . ; , ( , i (z). \Ire also have

W e agazn stress the fact that pseudo-natural solutzons sn the sense here dzs- cussed only exsst when the SSMP zs solvable. and these solutions are automatically solutions of the SShIP (not only of the SHhIP).

A simple convexity argument shows that every point in the lens-shaped region a, (z) n D, (z) is the value of the Stieltjes transform at z of a solution of the SSMP. It can be shown that all such values exactly cover thzs regzon, see [28].

4. Formulas of Christoffel-Darboux type

The determinant fornlula for the continued fraction (2.10)-(2.12) has the following form (see e.g.[26, p. 2301. [29, p. 111):

By multiplication and subtraction in the relations (2.10)-(2.12) with argument z and with argument ( we obtain the Christoffel-Darboux formula and associated forrnulas for orthogonal Laurent polynomials (see e.g. [26, p. 2311, [29, p. 111).

These formulas play a role in the theory of orthogonal Laurent polynomials and strong rnornent problems completely analogous to the role played by the classical Christoffel-Darboux formulas in the theory of orthogonal polynomials arid classical moment problems. In particular they are closely connected with the disk systems {A,, ( 2 ) ) as will be recalled in the next section.

iL7e shall in this section develop another system of formulas of the same general typc. iVe shall discuss in the next section how these formulas are connected with the disk systcms {D,, ( 2 ) ) .

Ebr brevity we sliall use thc following notation:

Taking into account (2.13)-(2.15) we may also write

It follows from (2.16)-(2.18) that all a t 1 ) are positive.

STRONG STIELTJES MOMENT PROBLEMS I ! THEOREM 4.1 Let z and < be complex numbers dzfferent from zero. Then the

followzng formulas hold for m = 1,2, . . . :

m-l

OLAV NJASTAD

PROOF: By multiplication and subtraction in the formulas (2.10)-(2.11) for the functions cp, and the arguments z and C we get

and

It follows from (4.11)-(4.12) that

Repeating this process and taking into account the initial conditions (2.12) we get (4.5).

In a similar way we get (4.6). By using analogous arguments involving functions $, as well as cp, and taking into account that (see (2.12))

we obtain (4.7)-(4.8). Finally by using arguments involving only the functions '$', we obtain (4.9)-(4.10). 0

The following result which is a consequence of Theorem 4.1 will be used in Section 5.


THEOREM 4.2. Let a,b,c,d be arbitrary complex numbers and let z , < be complex numbers different from zero. Then the following formulas hold for m = 1 , 2 , . . . :

[a$2rn(z) + b(P2m(z)][~$2m-l ( C ) + d ( ~ 2 m - I ( < ) I - [ a d ~ 2 r n l ( z ) + b ~ 2 r n - l ( z ) I [ ~ $ 2 r n ( C ) + d ~ 2 m ( < ) I 1

PROOF: By multiplying out on both sides of (4.15) and (4.16) and substituting from (4.5)-(4.6), (4.9)-(4.10) and (4.7)-(4.8) (the last also with z and < interchanged), the result follows. 0

5. Disks determined by pseudo-approximants

We recall that the disk systems {A, ( 2 ) ) are associated with the quasi-approximants, while the disk systems {D,, ( z ) ) are associated with the pseudo-approximants.

From standard properties of Moebius transformations it follows that the radius p,, ( z ) of An ( z ) is given by

The disk a,&) can be described by an inequality in the following way: n-1

w - w W E & ( z ) @ l@k(z) + w9k(z)12 5 -. z - z k=O

268 OLAV NJASTAD

Bu using formulas (4.1)-(4.2) and Christoffel-Darboux type formulas (see [26, p. 2311, 129. p. 111) we find that

71 - 1

It follows that the limit point situation for the system { A , ( z ) ) occurs if and only if the series CT=o lpk ( 2 ) l 2 diverges. For discussions of these properties of the disk system { & , ( z ) ) we refer to [16. p. 3731, [26, p. 3321, [29, p. 151.

Our aim in this section is to study the disk system { D , , ( z ) ) in some detail. Again by standard properties of Moebius transformations we find that the

center s , , ( z ) of D, , ( z ) is given by

while the radius r n ( z ) of D, , ( z ) is given by

THEOREM 5.1 The disk Dr , ( z ) can be described i n the following way:

n1- 1 U2k+l U 2 k + 2

U: E D2ni+l(z) 6 I f ? C - 1 + 2 k + l ( ~ ) + i~ i02k+ l ( zV k=O uZk+l u2k+2

(5.8) rn ulz - WZ

) @ 2 k ( ~ ) + ~ p 2 k ( ~ ) / ~ 5 -. z - z

k=O

PROOF: By solving the equation ui = -S,, ( z , T ) with respect to T we get

The point U: belongs to dD,, ( z ) if and only if 7 belongs to k. i.e., if and only if the imaginary part of the expression [ p , ( z ) d + v", ( z ) ] [p,, - I ( z ) ~ + & - I ( z ) ] is Zero. This means


By substituting from forrrlula (4.15) with n = 2m. a = c = 1, b = J, d = J. < = 2 we find that

and similarly with n = 2m + 1.

Taking into account (4.3)-(4.4) we find that dDz,,, (z) is given by the equation

and that 8D2,,,+l (z) is given by the equation

(Recall that the coefficlcnts in the orthogonal Laurcnt pol! nornials are real.) Since D,, (2) is contained in UJ and hence J = 0 does not hdong to Dl, (2). we conclude that DLllr (z) and (z) arc given by the inequalities (5.7)-(5.8). 0

IVe recall that all the coefficients in (5.7)-(5.8) are positive. Hence we may conclude also from thcse inequalities that the system {D,, (2)) is nested. LfTe shall now give an expression for the radius r,, ( z ) .

THEOREM 5 . 2 . The rt~dzus r,, (z) of Dl, (2) zs gzven by

270 OLAV NJASTAD

PROOF: From (4.1)-(4.2) we find that the numerator (inside the absolute value 'LLn 1 sign) in (5.6) equals ---- . - . The denominator is given by (4.5)-(4.6) with < = 2. un-1 z

Substituting these expressions in formula (5.6) for r,,(z) we obtain (5.15)-(5.16).

THEOREM 5.3. T h e l imi t ing disk D,(z) reduces t o a single point zf and only if the ser ies

diverges.

PROOF: The limit point case occurs if and only if lim r , = 0. Hence the n + x

result follows from Theorem 5.2. 0

6. Strong Stieltjes moment problems

We recall that the SHMP is determinate exactly when A, (z) reduces to a single point, and that this is the case if and only if the series xz==o /p,, (z)I2 diverges (for one or for all z in U). Our aim in this section is to discuss determinacy of the SSMP.

We note that this problem is determinate if and only if a, (z) n n, (z) reduces to a single point for all z E U. The following result shows that this can happen only if at least one of the disks A,(z) and D,(z) reduces to a single point.

This result has some analogy with a determinacy result for the classical case, where certain lens-shaped regions occur in the arguments. These lens-shaped regions are obtained by an approach different from that used here. For these results, in a more general setting, we refer to [ll], [36]. In [28] we show that the disks ~ , ( z ) used here can be obtained in a way analogous to that of the classical situation.

THEOREM 6.1. A s s u m e that t h e SHMP i s inde termina te . T h e n the SSMP is de termina te if and on ly if D,(z) reduces t o a single point for all z E U.

PROOF: First assume that the SSMP is indeterminate. It follows from (3.28) and the fact that the arc R,(z) does not reduce to a single point that ~ , ( z ) does not reduce to a single point.


Next assume that the SSMP is determinate. We find that in general

Here the absolute value of the first term to the left is l / r , , ( 2 ) . while for even n the absolute value of the second term to the left is l / p , , ( z ) . (A similar formula with odd n may also be used.) Since we have assumed that the SHhlP is indeterminate. it follows that l / p n ( z ) is bounded for each z in UJ. Since we have also assumed

Y ( 2 ) that the SSMP is determinate, it follows that the sequence {'1--) converges for %I ( ' )

every z E UJ (see [18, p. 5151, [24, p. 3381). and consequently the right-hand side of (6.1) is unbounded for every z E U. We conclude that l l r , , ( z ) is unbounded, which implies that lim r,, ( 2 ) = 0. This means that D,(z) reduces to a single point.

1 L " X

which completes the proof.

COROLLARY 6.2 Assume that the SHMP is indeterminate. Then the SSMP is determinate if and onlg if

for all z in U. PROOF: This follows immediately from Theorem 5.3 and Theorem 6.1. O

COROLLARY 6.3 Assume that the SHMP is indeterminate and the SSMP de- Vll

terminate. Then the sequence {-) i s unbounded. 1L71

PROOF: This follows immediately from Corollary 6.2 and the fact that the X

SHMP is indeterminate if and only if C Jp,, ( z ) l L < cm. 0 n=1

For the sake of completeness we shall also state a general necessary arid sufficient condition for the SSMP to be determinate.

We define polynomials P, , Q , by

zV'$ct.)nt(z) A n l ( z ) =

Lnr 277-2 ' ' ' $2 7 '

272 OLAV NJASTAD

By substitution in the recurrence formulas (2.10)-(2.12) we find that {P,) and {Q,,) satisfy the recurrence relation

where

The polvnomials P,, znd Q,, are thus canonical numerators and denominators -

- '"(') , and since the of a positive T-fraction (see [5]-[6]. [17]-[18]). Since "-- - - Qn(z) ~ n ( z )

@ (2) SShZP is determinate if and only if {"--) converges outside the non-negative real ~n (2)

axis. it follows that the SShlP is determinate if and only if the positive T-fraction x X

converges. This is the case if and only if at least one of the series C d,, , C en n=l 72=1

diverges. (See 117. p. 1461, [18, p. 5151). Consequently, by substituting from (2.13)- (2.15) in (6.8)-(6.9), we obtain a characterization of determinacy:

THEOREM 6.4. The SSMP is determinate if and only if at least one of the following series dzverges:

COROLLARY 6.5 The following two statements are equivalent:

(A) At least one of the series (6.10) - (6.13) dzverges. X

( B ) A t least one of the two serzes (5.19) and C lp,,(z)/2 dzverges for all z € UJ 17 =I)


PROOF: This follows from Corollary 6.2, Theorem 6.4 and the fact that the s

SHMP is determinate if and only if C lvn(z)12 = m. 0 n= 1

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in Analysis, Hafner, New York, 1965. 2. C. Berg, Markov's theorem revisited, J . Approximation Theory 78 (1994)

260-275. 3. C. Berg, In,determinate moment problems and the theory of entire functions,

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5. C.M. Bonan-Hamada, W.B. Jones and 0 . Nj6stad, Natural solutions of indeterminate strong Stieltjes moment problems derived from PC-fractions, In: Orthogonal Fun~t~ions, Moment Theory and Continued Fractions: Theory and Applications, (W.B. Jones and A. Sri Ranga, editors), Marcel Dekker, 1998, pp. 15-30.

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7. T.S. Chihara, On indeterminate Hamburger moment problems, Pacific J . Math. 27 (1968), 475-484.

8. T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.

9. T.S. Chihara, Indeterminate symmetric moment problems, J . Math. Anal. Appl. 85 (1982) 331-346.

10. L. Cochran and S. Clement Cooper, Orthogonal Laurent polynomials on the real line, In: Continued Fractions and Orthogonal Functions, (S. Clement Cooper and W.J. Thron, eds.), Marcel Dekker, New York, 1994. pp. 47-100.

11. V.A. Fil'stinskii, The power moment problem on the entire axis with a finite number of empty intervals in the spectrum, Zap. hlekh. Mat. Fak. Khar'kov. Gos. Univ. Khar'kov Mat. Obshch 30 Ser 4 (1964) 186-200 (in Russian).

12. P. Gonzalez-Vera and 0 . Njbstad, Convergence of two-point Pad4 approximants to series of Stieltjes, J . Comp. Appl. Math. 32 (1990) 97-105.

13. H. Hamburger, ~ b e r eine Erweiterung des Stieltjesschen Momentproblems, Parts I, 11, 111, Math. Annalen 81 (1920), 235-319; 82 (1921), 120-164; 82 (1921), 168-187.

14. E. Hendriksen and H. van Rossum, Orthogonal Laurent polynomials, Indag, Math. 89 (1986). 17-36.

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15. W.B. Jones. 0. N j k a d , and W.J. Thron, Orthogonal Laurent polynomials and the strong Humburger moment problem, J . hlath. Anal. Appl. 98 (1984), 528-554.

16. \V.B. Jones, 0. Njast,ad and W.J. Thron, Continued fractions and strong Hamburger momen,t problems. Proc. Lond. Math. Soc. 47 (1983), 363-384.

17. \V.B. Jones and W. J. Thron. Continued Fractions: Analytic Theory and Ap- placations. Encyclopedia of Mathematics and its Applications 11 Addison- \Vesley (1980).

18. 1tT.B. Jones. W.J. Thron and H. \aadeland, A strong Stieltjes rrwrrrent problem, Trans. Amer. Math. Soc. 261 (1980), 503-528.

19. H.J. Landau (ed.), Moments i n Mathematics, Proc. Symposium Appl. Math. 37. Arnrr. Math. Soc.. 1987.

20. R. Nevanlinria. Asymptotische Entwicklungen beschrunkter Funktionen und das Stieltjessche Momentproblerns, Ann. Acad. Sci. Fenn. (A) 18 (1922).

21. R. Nevanlinna, ljber beschriinkte analytische Funktionen, Ann. Acad. Sci. Fenn. (A) 32 (1929) 1-75.

22. 0 . NjBst,ad, Solutionu of the Stieltjes moment problem, Communications in t,he Analytic Theory of Continued Ractions 3 (1994), 32-51.

23. 0. Njastad. Classical and strong moment problems, Communicatio~ls in the Analytic Theory of Continued Fractions 4 (1995): 4-38.

24. 0. Njastad, Solutions of the strong Stieltjes moment problem, Methods and Applications of Analysis 2 (1995), 320-347.

25. 0. Njastad. Extremal solutions of the strong Stieltjes moment problem,. J . Cornp. Appl. hlath. 65 (1995), 309-318.

26. 0. Njastad, So1ution.s of th,e strong Hamburger moment problem, J . Math. Anal. Appl. 197 (1596). 227-248.

27. 0. Njastad. Remark., on canonical solutions of strong moment problems. In: Orthogonal Funct,ions. illoment Theory and Continued Fractions: Theory and Applications. (W.B. Jones and A. Sri Ranga, editors), Marcel Dekker, 1998. p p 359-374.

28. 0. Njastad, Lens-shaped regions for strong Stieltjes moment problems, Meth- ods and Applications of Analysis, to appear.

29. 0. Njastad and W.J. Thron, Unique solvabilzty of the strong Hamburger moment problem, J . Austral. Math. Soc. (Series A) 40 (1986); 5-19.

30. I-I.L. Pedcrsen: Stieltjes moment problems and the Friedrichs extension of a posztzve deJnite operator, J . Approx. Theory 83 (1995) 289-307.

31. H.L. Pedersen, La pararne'trization de Nevanlinna et le problkme des mo- in,en,ts de Stieltjes inde'termine', Expo. Math. 15 (1997) 273-278.

32. 0. Perron, Die Lehre von den Kettenbriichen. Vol. 2 (3. edition), Teubner (1957).

33. hl. Riesz, Sur le problkme des moments, I , I I . III, Arkiv fiir hlathematik, Astronomi och Fysik 16 (12) (1922); 16 (19) (1922): 17 (16) (1923).

34. J.A. Shohat arid J.D. Tamarkin, The Problem of Moments, Mathematical Surveys No. 1, Anler. Math. Soc., Providence. R..I., 1943.


35. T.J. Stieltjes. Rech,erches sur les fractions continues. Aim. Fac. Sci. Toulouse 8 (l894), J . 1-122, 9 (1894), A 1-47; Oeuveres 2, 402-566.

36. K.I. ~vecov, On Hamburger's momen t problem with the supplementary re- quirement that masses are absent on a given interval. Cornnxln. Soc. Alat,h. Kharkov 16 (1939) 121-128 (in Russian).

37. W. Van Assche, The ,impact of Stieltjes' umrk o n con t i ' n~~ed f rnc t~ons and orthogonal polynornials. In: T.J. Stieltjes. Collected Papers. Springer-V~-lag 1993.

DEP~\RTSIENT O F . \ IA'I 'HE~I~\TIc~\I , SCIENCES, ~ O R \ Z ' E G I A N I;XI\'EI<SII.Y OF S C I E N C E .\XI) ~'I.:('H-

NOLOGY, K-7031 T R O X U H E I ~ I , KOH\VAY E-mad address: n j astadQmath . ntnu. no

Contemporary Mathematics Vnlump 236. 1999

Weak asymptotics of orthogonal polynomials on the support of the measure of orthogonality and considerations on

functions of the second kind

Franz Peherstorfer. Robert, Steinbauer

ABSTRACT. In a previous paper 1141 the authors studied perturbations of the reflection coefficients of orthogonal polynomials on the unit circle and derived strong ratio asymptotics on the closed unit disk but outside the support of the measure of orthogonality. Here we study the effects of such perturbations on the support of the orthogonality measure. We will derive weak asymptotics and show that this kind of convergence cannot be improved in general. These results are used to obtain strong comparative asymptotics of the related functions of the second kind. On the one hand, the functions of the second kind are very useful in the explicit description of (ratio) asymptotics of orthogorial polynomials and on the other hand they are a main tool for the function- theoretical approach to the study of orthogonal polynomials on the unit circle, since they are very closely related to Carathbodory- and Schur functions.

1. Introduction

Let M be the set of all nonnegative finite Bore1 measures a on [O, 271.). normalized by

It is well known that there exists a uniquely determined sequence of morlic orthogonal polynomials {P,,(z, a ) = zrL + . . . )rz0 on the unit circle with respect to a. that means

where the value d, ( a ) is given by

1991 Mathematics Subject Classzfication. 42C05. Key words and phrases. Orthogonal polynomials on the unit circle; furictions of the second

kind, weak convergence, ratio asymptotics. This work was supported by the Austrian Forids zur Forderung der wissenschaftlichen

Forschung, project-number P12985-TEC.

278 FRANZ PEHERSTORFER AND ROBERT STEINBAUER

Often it is more natural to deal with orthonormal polynomials a,, (z, a) instead of the monic ortliogonal polynornials. which satisfy

kkom (1.1) we irnmediately get that

hlonic orthogonal polyriornials P,, (z , a) on the unit circle can be completely described by a sequence of complex numbers, the so called reflection coefficients. That is. the Pl,'s satisfy a recurrence relation of the form

where, P:(z, u ) := ~ ~ ~ P , , ( l / z , a) denotes the reversed polynomial. It is easy to see that

i.e., the reflection coefficients are the evaluation of the monic orthogonal polynomials at the point z = 0. Furthermore. they satisfy

(1.5) la,,(a)l < 1 for a1ln.E NO.

In this papcr we study two sequences of ortlionormal polynon~ials {@,,(z, a,))) and {(a,, (z, a ) ) . where we suppose that the second sequence results from the first one by a compact perturbation of the reflection coefficients, i.e., we assume that

If this convergence is sufficiently fast. say 7:

I t =O

then we derived comparative asyrnptotics of the form lim,,,, (z, a)/@,*, (z, ao) in tht, interior of the unit circle as well as on the unit circle outside the support of the measures of orthogonality [14]. In Section 2 we study the question what's happening on thc support of orthogonality. We will prove weak convergence of the orthogonal poly~loniials on the support and we shall show that in general this type of convergence cannot be improved. In Section 3 we deal with asyrnptotics of functions of the sccond kind, where these functions correspond with the measure of orthogonality in a unique way and are closely related to Carathkodory- and Schur functions, which, in their turn. are the main tool for a functional theoretical approach to orthogonal polynomials.

2. Weak asymptotics on the support of the orthogonality measure

In this section let {a,, ( 2 , ao));l",O be a given sequence of orthonormal polynomials. We are interested in the question how these polynomials are related to a ncw sequerlce of orthonorn~al polynomials {@,,(z, a))$=,, on the support of the measure of orthogonality. whcre the new polynomials arise from the original ones by a compact perturbation of the reflection coefficients. that is

ORTHOGONAL POLYNOMIALS AND FUNCTIONS O F T H E SECOND KIND 279

In [14] the authors studied ratio asymptotics of the perturbed and unperturbed orthogonal polynomials outside the supports of the orthogonality measures.

To state and prove our results, let us consider some basic facts which hold true for all orthogonal polynomials on the unit circle: Let p E M be a given measure. From (1.3) and the relation

see e.g. [2, Theorem 1.81, one can derive that

for all n E N, compare [7, formula (6)]. Let us consider the infinite matrix (compare again [7, p.4011)

Then we can express relation (2.2) in the compact form

and as a consequence

z k u ( z , , u ) = U ~ ( ~ ) U ( Z , ~ ) for a11 k E N

Notice that Uk((CL is a triangle matrix with ( k + 1) superdiagonals. Let us denote

U k ( P I = (&(cL)):~=,,, then

and yet

With the help of this structure we can derive:


THEOREM 1. Let a0 and a be orthogona,lzty measures from M related by (2.1). Then for all j E Z the following limit relation holds:

where 5 denotes the weak* limit on [0,27r), that is

for all bounded measurable and Riemann-integrable functions g

Proof. Let us first show that

(2.6) lirn (lT ezkpQrz(eZ*', a)@,,+, (ez*', a) da (p ) n--'x

holds for all k , g E Z. We start with the case that k E No. By (2.3) we have to show that

k lim (ufi, ,+, (a) - U , n + ~ (00)) = 0 I1 ' X

for all fixed J E Z. It is not difficult to see that by the triangle structure of the matrix U the elements on the diagonals {u: ,,+, (p ) )~x7 , , I , p E {go, a), are all generated by the same law in terms of a finite number (depending on j and k) of reflection coefficients, i.e.. u?,,+,) (71+1)+J(p) results from uk ,+,(p) by a 1-shift of the indices of the reflection coefficients. Now recall that a,,(p) = P,,+, (0, p ) and

Then (2.6) follows from (2.1). For negative k one only has to take complex conju- gation in (2.6).

Hence, we have shown the desired limit relation (2.4) resp. (2.5) for all functions g of the form g(p) = e " ~ , where k is an integer. By linearity and continuity (recall also the Banach-Steinhaus Theorem) relation (2.5) is also true for an arbitrary continuous 2~-periodic function. Applying one side approximation arguments (see e.g. [20, Theorem 1.5.4]), the assertion follows. 0

Let us note that the limit relation (2.4) (resp. (2.5)) holds true if one replaces [O, 2 ~ ) by any subinterval [dl , &], because one can choose the function g to vanish identically outside the interval [dl , d2].

Especially, if we put j = 0 in Theorem 1, then we get

If the reflection coefficients P,(O, a) and P,, (0, ao) tend to zero, wen more is known: Let us first consider the Szego-class, that is the class of all finite and nonnegative

ORTHOGONAL POLYNOMIALS AND FUNCTIONS O F THE SECOND KIND 281

Borel-measures p on [ O , ~ T ) which satisfy

or equivalently .x

One of the main results of Szego's theory is the limit relation (see [20, Chap. X and XI, pp. 274-2951 and [2, Chap.V, pp.187-2721)

where the so-called Szego-function D ( z , p) is given by

and where t'he boundary values satisfy

for almost every p E [O,27r). Hence, if we assume both a 0 and a from the Szego-class, one obtains from (2.8)

lim lI@,(eZp, a)D(e2@, a) - Qn(eZp, ao)D(ez+, 00)ll L > ( [ o 2 ~ ) ) = 0, n-3i

that is convergence in L2([0, 2~))-sense. In the case that only p1 > 0 almost everywhere in [0, 27r), then

lim Pn (0, p) = 0 n - x

and M&tk, Nevai, and Totik [8, Theorem 2.11 have shown that

Again, if both a:, > 0 and a' > 0 almost everywhere in [0, 27r) then

If the reflection coefficients Pn(O, 00) do not tend to zero, then in contrast to the case just discussed, the limit relation (2.7), and hence the results of Theorem 1. cannot be improved in general, neither by replacing the weak-* limit by LI-convergence nor by pointwise convergence almost everywhere. We will demonstrate this fact by the following

Counterexample. Consider the two sequences of reflection coefficients

where, for simplicity, a E ( -1 , l ) \ (0). It is known. see e.g. [3, 7, 151. t,hat go

and a are absolutely continuous on [a, 27r - a], with cos a / 2 := ,/-. The


Radon-Nykodym derivatives ah and a/ are given explicitly by

for cp E [a, 27r -

and

a[)(cp) = al(cp) = 0 for cp E [O,27r) \ [a, 27r - a] .

For the representation of f we have used [15, formula (2.21) and Theorem 4.31. The value a can be chosen such that a0 is an absolutely continuous measure and that a has at most two mass-points at * P , where /3 E (0, a); a possible choice for instance is a = 0.6.

We will show that

(2.11) (1@71(ezp,ai12f (9 ) - ao)12fo(cp)) % O 71 - X

pointwise almost everywhere on [a, 27r - a].

Let us remark that it makes sense to consider convergence only on [a, 27r - a] and not on [O,27r) because supp(ao) \ [a, 27r - a] = 0 and @, (e*@, a) --- 0 as n + CQ.

It sufficies only to show (2.9) because then, as a consequence,

from which (2.10) follows. For this last implication we made use of the fact that the a, (e", ao)'s and a,, (ezp, 0) 's are uniformly bounded away both from zero and from infinity on [a + c,27r - a - €1, for all 6 > 0, which can be seen from [16, Corollary 2.2 resp. Lemma 3.11.

Moreover, the left hand side in (2.11) cannot converge to zero almost everywhere, because otherwise, by the equi-integrable property of the orthonormal polynomials and by standard results from measure theory, this would imply that also the L1-limit in (2.10) would be zero.

Let us now prove (2.9): It can be derived from [17, Corollary 11 that

for every measurable and Riemann integrable function g. Further, recall the fact that for every measure ,LL E M by [2 , Theorem 2.21

Since IQ7,(eV. o)I2 is a trigonometric polynomial of degree n we have

ORTHOGONAL POLYNOMIALS AND FUNCTIONS OF THE SECOND KIND 283

Now, if the left hand side in (2.9) would converge t o 1 in the weak-* sense then

for every measurable and Riemann integrable function g. Take

fo(p) s i n $ - n . s i n % - g(p) := -- - sin 5 , c p E [ % 2 ~ - - - 1 ,

f ( 9 )

then

and by (2.14)

For the first integral at the right hand side we can write

sin 5 by (2.13)

27;-o sin 5 - a . sin * dp. S , ..-

If we divide the identity

(see [12, Corollary 3.11) by P,+, (z, a(]) and if we use the limit relations, given in [16. Theorem 2.11. we arrive at

uniformly compact on C \ { e L p : y E [a, 2~ - a ] } , where


Hence: we get for the left hand side in (2.15)

But this value is in general different from the right hand side of (2.15), which has the form

2.rr-tu . sln 5 - a . sin Q c, := S, "$0.

sin 5 To see this: we calculate both values cl and c 2 with a = 0.6 (compare above). Using Mathematica, we get

cl = 4.80319 # 3.78767 = CZ.

3. Functions of the second kind

Using the results from the previous section we will now derive asymptotics for the functions of the second kind corresponding with an orthogonality measure on the unit circle.

First. we will need some further notation: Let { a , ( p ) ) L be a sequence of reflection coefficients, p E M, then the monic polynomials of the second kind {On (2, p))F==o are given recursively by

f l n + i ( z , p ) : = z f l n ( z , p ) - a , ( ~ ) f l ~ ( z , ~ ) , ~ E N o , O O ( Z , P ) = ~ .

The orthonormalized polynomials are denoted by

Further, let

be the Carathkodory function related to the measure p. There is a one-to-one correspondence between the Carathkodory function F ( z , p ) and the measure p , which can be rediscovered from F ( z , p ) by the inversion formula (see e.g. [6, formula (11.7): p.171)

(3.2) '(li + + '(' - = const. + lim Re F (re". , p) d$.

2 r-1-0

Finally. we can state the following Definition. Let p be a given measure from M, then

121 < 1. n E No, denote the functions of the second kind


Let us demonstrate some applications of functions of the second kind: (i) Approximation of Carathe'odory functions: Obviously

which is by [6, Theoorem 13.11 of order O(z7"') for J z / < 1. Hence, the functions of the second kind can be considered as a measure how well the Carathkodory-function F (z , p ) can be approximated by the ratio PFL (2, p)/@: (z, p) .

(ii) Associated measures: For a given measure p from M we can define the u-th associated measure p("), which is related to the shifted reflection coefficients

a 7 , ( ~ ( y ) ) := aTL+"(p), n E NO.

In view of [12, Theorem 3.11 we have

Thus, if one wants to know the asymptotic behaviour of the u-th associated polynomials @, (z, p") with respect to n as well as with respect to u one should know about the behaviour of the functions of the second kind (see [14, 161).

(iii) Schur functions: Let

be the Schur function corresponding with the Carathkodory-function F(z , p) . Schur functions, i.e., analytic functions on {lzl < l} with sup{l f(z) l : lzl < 1) 5 1. have been extensively studied by Schur in [18, 191. Using Schur's algorithm, see e.g. [6, Theorem 18.1 and Theorem 18.21. which is given recursively by

one can discover the reflection coefficients via the relation

f"(O! P) = ~ " ( c L ) .

From [6, formula (18.10), p.341 we have the explicit representation

Furthermore, all f,'s are again Schur functions, that is f,, analytic and 1 f,(z)l < 1 on {(z / < I ) , and by (3.4) and (3.7) there holds

i.e.,

f1,(z, P) = f (z, p i ' ) ) . (compare also [6, Theorem 18.2, formula (18.8)]).

(iv) Connectzon wzth functzons of the second kznd on the real h e : It is known and important for asymptotic investigations that polynomials orthogonal on [-I, 11


can be expressed with the help of polynomials orthogonal on the unit circle. It has been shown by the first author [ll, Section 31 that there is a corresponding relation between functions of the second kind and how to get asymptotics with the help of this relation.

(v) Stieltjes-polynomials and Gauss-Kronrod quadrature: Stieltjes-polynomials are defined as partial sums of the reciprocal of the function of the second kind. Roughly speaking, to get asymptotics for Stieltjes-polynomials means to derive asymptotics for the functions of the second kind (see [ll, Section 41).

Note that the functions F ( z , p) , &(z, p) and E,(z, p) are elements of the Hardy spaces H, = H,({lzl < I)) , p < 1 (see e.g. [l, Theorem 3.21). But if we take the integral representation in (3.1) and (3.3), we can consider the functions F ( z , p) , 6, (2, p) and 'Hn (2, p) to be defined on @ \ {e" : p cp supp(p)), where supp(p) denotes the support of the measure p. Then it is not difficult to obtain the following relations

for all z E @ \ {e" : p E supp(p)). Furthermore, the estimates

(this can be easily derived from [6, formula (12.13), p.201 by setting ~j = F ( z , p ) ) and

(follows directly from the integral representation or from (3.8)) hold true. From the well known relation (see [6, formula (5.6), p.71)

one can derive the identity

for all n E No and for all z E @ \ {eLq : p E supp(p)). Let us state some further basic properties:

L E M M A 1. Let K be a compact subset o f : \ {e'p : p E supp(p)), p E M. Then there exists a constant M = M ( K , p ) < OC, independent of n, such that

for all n E N and z E K . Furthermore, for all n E N


where equality is only possible if lzl = 1, and

Proof. The following identity holds

1 2"

= ( ) ( Z ) + 1 (icn(l')ezTfP+ terms of lower order)@,,(eZc, P) ~ P ( P ) 0

Since K has a positive distance to the set {eZq : p E supp(p)) - note that supp(p) is a closed set - we obtain that

Hence,

and the uniform boundedness of and z@,'H, follows. From (3.9) and the well known fact that /@,,(z ,p) / < I@,*,(z,p)J on {lzl < 1)

and j@n(e",p)I = I@,*L(eZP,p)! we get the estimate

and the identity (3.11) is only possible if I@;L(z)Gr,(z) 1 > 1 on K n {lzl 5 1) (where we have strict > 1 on { Jz J < 1)). The analog statement on z@,,'HT, follows from (3.8).

Finally, the limit relation

2 x e" + z 1 27I --- /@n(e'",p)12 +(P) = -I;; 1 @,.(ezP. P) I '~P(P) = -1 lim -

z - i x 27l eZ+ - z 0

and the representation (3.15) gives the last assertion. Remark. From Lemma 1 we see that


Especially. if p is not in Szego's class, i.e.,

X

log p' @ L1 ([o, 27r)) or equivalently la,, (p) 1" 00,

n=O

then bl- [6, Theorem 21.11

(3.16) lim ~ Q : ( Z , ~ ) = m uniformly on {lzl 5 r < 1) 1 1 - X

and. thus.

(3.17) lim G,, (z, p ) = 0 uniformly on { I z / I r < 1). 7L'X

Besides. condition (3.16), and hence also condition (3.17), is also necessary for a measure p not to belong to Szego's class.

If, on the other hand. p is in Szego's class, then it is well known that (see e.g.

12, 4, 6, 10, 201)

lim Q:,(z.p) = D(Z, uniformly on {Izl < r < 1) 7 1 - X

and it is shown by the first author in [ll, Theorem 2.11 that

linl G,, (z. p ) = 2 . D(z, p') uniformly on {lzl < r < 1). TI - X

Herr D(z, pl) denotes the Szegii-function defined by

el*? + z el*? - z

But surprisingly we have in any case. by [ll, Theorem 2.11,

(3.18) lirn X,, (z, p ) = 0 uniformly on {lzl < r < 1). 11 - x

Having Lemma 1 in mind, we can prove the following asymptotic relations.

PROPOSITION 1. Let the measures oo,o E M be related by (2.1). Then

uniformly on Furtherm

A. i A and

compact suhsefs of @ \ {el+' : p E supp(oo) U supp(o) ) . ore, for every znfinzte zndex-set A C No there exzsts a n znfinzte subset a functzon O(Z) = O(z;o ,Ao) , whzch 1s analytzc on @ \ (ez" : p E

supp(o)) and which has no zeros outside the unit circle. such that

(3.22) lim za,, (z. o)'H,, (z, o ) = O(Z: rr, A()) - 1 n - x 11 t \ , I

unzforrnly on compact subsets of C \ {e l9 : p E supp(a))


Proof. Let z E @ \ {eZ9 : p E supp(uo) U supp(a)) be fixed. Then the limit relations in (3.19) follow (pointwise) from (3.15) and (2.4) in Theorem 1. Since all the functions are uniformly bounded on compact subsets of @ \ {e" : p

supp(ao) U supp(a)), recall Lemma 1, the uniform compact convergence holds by Vitali's Theorem.

Next, let us define the functions

Now consider the sequence of distribution functions (resp. measures) { ~ , ( p ) ) , ~ , ~ given by

This sequence is uniformly bounded and by applying Helly's Theorem we can extract a subsequence {an)nEn,, A0 C A, which converges in the weak-* sense to a distribution function a, := a,(Ao). Hence, the function

2T ezp + z O(Z; a, Ao) := lim O,(z. a) = & e"-z n - x

d u x (PI

exists for all z E C \ {e2+ : p E supp(a)) and is analytic there. Further, using the notation z = reZa we have

and one obtains that O(z) # O on {izl # 1). The limit relations (3.21) again follow from (3.15). 0

Let us recall the above mentioned relations of Carathkodory-, Schur- and functions of the second kind

The next theorem gives asymptotics of the n-th associated Carathkodory and Schur functions.

THEOREM 2. Let the measures a(), a E M be related by (2.1). Further let S c (121 = 1) be the closure of all zeros of G,(z,P) - zXn(z, P), where ,LL E {a(), a) and n E No. Then

uniformly on closed subsets of C \ ( { e i p : cp 6 supp(ao) U supp(a)) U 5') and

'Note that each a,, is monotone increasing and a,,(2n) = 1


unzformly on cornpact subsets of {lzl 5 1 ) \ ({eLq : p E supp(ao) U supp(a)) U S ) .

P~oof . By (3.23) the first limit relation can be written as

Since, by (3.8). F ( z , p ) = -F( l /z , p) for every orthogonality measure p it sufficies to show convergence only on a compact set K C {lzl < 1) \ ({eL9 : p E supp(a0) U

supp(a)) u S). In a first step we will prove pointwise convergence and uniform boundedness

on K. Then Vitali's Theorem will imply uniform convergence on K. Lvt i1 E {ag, a ) . The uniform boundedness of F ( z , p ) on K can be obtained

from the intcgral representation (3.1) and from the following relation

(3.26) s u p p ( p ( ~ ) ) C supp(p) U {p : el9 E S )

To see (3.26). let us recall that by (3.4)

and let 1V be a closed subset of [O,27r) \ (supp(p) U {p : el' E S)). Then it is not difficult to derive from (3.27) that

~e ~ ( e ' ~ , p ( " ) ) = lirn ~e re", p('") = 0 for p E N 7 -1

and from the inversions formula (3.2) relation (3.26) follows. Now let z E K be fixed. We apply [14, Lemma 3.11 and together with [6,

Theorem 13.11 we obtain

where 5,,(2) is indepe~ldent of n and 6,(z) + 0 as v + m (compare also the proof of Lenlrna 2 in [14]).

Let us now rnention the following estimate. which holds on the whole closed unit disk {IzI < 1):

It is easy to see from (1.3) by using an induction argument that

where the constant c,, only depends on v. Thus, for every fixed v the limit relations


hold. Using the integral representation of the CarathPodoy function ~ ( z , ah")) i l l ) we see from (3.26) and (3.28) that the sequence {9: (2 . n,, ) /@:,(2, o ~ , ' ~ ) ) } , ~ ~ ~ ~ ~ ~ is

bounded. The sequence { l / @ t ( z , O ( " ) ) ) , , ~ N , , is bounded as well, as the following considerations show: If / z J < 1 then /@:(z , a ( " ) ) 1 2 Jw by [5. formula (1.12')] and if lzl = 1 then

( a t ( z , a ( ' ' ) ) p q z , a ( n ) ) 1 = 1 I - > const.,

where by [14, formula ((4)) in the proof of Lemma 3.1 and Remark S . l ( h ) ] the constant can be chosen independent of n and v .

Collecting all the above given informations we derive

lim 71 - X

) - 0

for every fixed v E N. Now. for a given t > 0, we choose uo = v c ~ ( c ) sufficiently large such that by

(3.28) for all n E No

f - F ( Z , ~ ~ ' ) ~ < - 3 and I *ill ( 2 ,

@tob, a ( " ' )

Then take no = n o ( € ) as large such that by (3.29) for all n > no

and (3.25) follows for every z E K. By the already proven uniform boundedness of the Carat hi.odory functions

F ( z , a,j")) and F ( z , a ( " ) ) on K and by Vitali's Theorem this is already the desired uniform convergence of (3.25) on K. This proves the first part of the tlicore~n.

From (3.9) and (3.10) we have

w e s for all measures p E M. This estimate and (3.11) 6'

Thus, we can write

F ( z , P ( ~ ) ) is a Carathkodory function and therefore Re F ( z , P ( " ) ) 2 0 oil K . Hence. we obtain for all z E K and n E No

Now by the first part of the theorem, i.e..


the assertion follows." 0

Remark. As already mentioned in the proof of Theorem 2, relation (3.24) is by (3.4) equivalent to the limit relation

lim ( ~ ( z , u p ) ) - F ( z , a("))) = 0 n-3:

which implies moreover (compare (3.29))

Let us point out once more that there are no assumptions on the reflection coefficients {a,(ao)) and {a, (a)) apart from (2.1).

From (3.18) and [14, Lemma 3.1 (a)] we know that

lim l-In(z, p ) = 0 uniformly on K, n - x

where the set K is given as in the proof of Theorem 2. Furthermore,

lim 6, ( z , p) = 0 uniformly on K n i x

if and only if p do not belong to Szego's class (to see uniform convergence in a neighbourhood of the unit circle use [14, Lemma 3.l(d)]).

The following corollary gives some result on the speed of these convergences.

COROLLARY 1. Let a be an orthogonality measure from M and let a E C with la1 < 1 be fixed. Then there holds

lim a n ( a ) = a n --t x

if and only if

( a ) 1 J(z - + 41aI2z + z ( l - 2a) - 1 (3.33) lim - - -

n-= G,(z, a) z J(z - + 4 / ~ / ~ z + (1 - 2a) - z

uniformly on {lzl < r < 1).

Proof. Let us assume that lim,,, an (a ) = a. Take {an(ao) := a)nEWo as a comparison sequence. Then by [15, Thm.4.4 resp. Thm.5.1 and Example 5.l(a). p.100]3

(n) (z - + 41aI2z - (a + az) F (z , go) = F ( z , o, ) = for all n E No.

(1 - a) - (1 - ti)z

By Theorem 2 we have

2 ~ o r the point z = 0 note that H n ( z , uo) /Gn ( z , uo ) = an (uo ) and Hn ( z , u) /Gn(t , u ) = a n ( u ) , i.e., we have pointwise convergence to zero of the difference. By (3.30) and Vitali's Theorem we obtain uniform convergence in a neighbourhood of z = 0 and, thus, on K.

"et us mention that we use here a slightly different notation than in 1151. One has to replace a in 1151 by -a.

ORTHOGONAL POLYNOMIALS AND FUNCTIONS O F T H E SECOND KIND 293

uniformly on {Izi < r < 1); recall that by (3.9) there holds Iz'H,,(z. a)/G,(z, a ) / < IzI for IzI < 1. NOW (3.33) follows quite easily.

Next suppose that (3.33) holds. We will apply this relation at the point z = 0. From (3.3) we obtain

and, hence,

Now, 1'Hospital's rule shows that the above right hand side is nothing but -a. This proves the assertion. 0

Remark. (a) Let us mention that a corresponding result as stated in Corollary 1 also holds true if we consider asymptotically N-periodic reflection coefficients a,,(a), i.e.,

(3.35) linl a 3 + , , ~ ( u ) = a3(a0) for j = 0 , . . . , N - 1, n+?r

where a 3 + 7 L ~ ( u ~ ) = a3(uo) for all n E N and j E (0, . . . , N - 1). Indeed, it can be shown with the help of the relations

note that by the N-periodicity of the a,(ao)'s the equation F ( z , a:;'+7Lx)) =

F ( z , a t ) ) is valid, that (3.35) holds if and only if

for j = 0, . . . , N - 1. Recall that by (3.4) F (z , a:;)) and by [15. pp. 97-99] F (z , DO)

are known. (b) For the Nevai class, i.e., for the case that a,(a) + 0, Corollary 1 has the nice

form: lirn,,, a,(a) = 0 if and only if lim,,, 'H, (z, a)/G,, (z, a ) = 0 uniformly on (1.1 5 7- < 1).

For the rest of this section we always assume that

lim sup la, (00) I < 1 n - x

(otherwise a 0 would be a singular measure). Further we consider more restrictive assumptions on the perturbation of a given sequence of reflection coefficients {a,(ao)): in what follows we we suppose instead of (2.1) that

Under these assumptions we have shown the following asymptotics of t,he corresponding orthonormal polynomials (see also our survey paper [13]):

THEOREM 3. ([14, Theorem 2.2 and 3.21) Let a 0 and u be given orthogonal measures from M which satisfy (3.36) and (3.37). Then there hold:


(a) Let K1 be a closed subset of {Izl < 1) and assume that the sequence

where t,, := la,(ao) - a,(a) 1, then

lim

uniformly on K I . Here, the analytic function A is the uniform limit

(b) Let K2 be a closed subset of {lzl = I), which has a positive distance to supp(p) and to the zeros of 4, (z, p ) * z'H, (2, p ) , p E {ao, a ) . Further assume that instead of (3.38) the sequence

(3.40) CL) I@:(z, m d 2 is uniformly bounded on K2 { I @ ; L ~ , a ( ) ) 1 2 }ncNo

Then the limit relation (3.39) holds uniformly on K2 .

With the help of Theorem 3 and the above results we can derive ratio asymptotics for the functions of the second kind.

THEOREM 4. Let 00 and a be given orthogonality measures from M which satisfy (3.36) and (3.37) and let K be a closed set i n :, which has a positive distance to supp(p) and to the zeros of 4,,(z, p ) * z'H, (z, p ) , p E {no, a ) . Further suppose that (3.38) holds i n the interior of the unit circle and (3.40) o n a set K n (1 - t 5 lzl 5 l), E > 0 sufJiciently small, if this set is nonempty. Then

lim = ~ ( z ) on K n {lzl 5 1) n-x 4n(z, 0)

lim "n(z'"" = A(z) on K n {lzl 2 I), T ~ + X 'H,(z,a)

where both limit relations hold uniformly on compact subsets ( in c). Proof. From the integral representation (3.3) of G,(z,p), p E {ao,a), it is

easily to see that

Now, Theorem 1 and (3.12) in Lemma 1 give

lim @:L(z, ao)G,(z, ao) = 1 uniformly compact on K n { lzl 5 1).

I , - - @;(z, a)GTL(z, a)

The first limit assertion follows from Theorem 3. The second one follows from the first one by taking into consideration that

and by (3.8)


In general we will not have convergence of 'Hn(z, oo)/3-1,(z, a ) on {Jz / < 1) as the following example shows: Let a be the Lebesgue measure on [O, 27r). Then a,(a) = 0 for all n, a,, (z, a ) = Q,,(z, a ) = zn, F (z , a ) = 1 and, consequently, %,(z, a) = 0 on {lzl < 1) for all n E No. If 3-1,(z, ao) $ 0 identically, then 3-1, (z, ao)/3-1, (z, a ) cannot converge in C.

But we can say the following:

COROLLARY 2. Let the assumptions of Theorem 4 be fulfilled and put k,(z) :=

zan(z , o)XFtn(z, a()) for z E K . Then

lim

uniformly on K1 := K n {lzl 5 1).

Proof. Let us write

where we used (3.11) for the last identity. Taking the lirnit TL + m at both sides, we obtain by Theorem 4

uniformly on K1. By [14, Lemma 2.21 the function A has no zeros on KI" and is analytic there. Hence, (3.41) reduces to

uniformly on K 1 . This is the assertion. 0

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(Peherstorfer) INSTITUT FUR ~IATHEMATIK. JOHANNES KEPLER UNIVERSIT~T LINZ. d0d0 LINZ- AUHOF, AUSTRIA

E-mail, Peherstorfer: [email protected]

(Steinbauer) INSTIT~JT t UR ~IATHEMATIK, JOHANNES KEPLER UNIVERSITAT LINZ. 1040 LINZ- AUHOF. AUSTRIA

E-mail, Steinbauer: [email protected]

Contemporary Matherrlatics Volume 236. 1999

Trees of Approximation Constants

Serge Perrine

This text is dedicated t o Christzan Pierret.

ABSTRACT. The article recalls first the formalism which is necessary t o generalize the classical Markoff theory, and the known results concerning the diophantine equation:

zL + tJ2 + z2 = 4 x y z - z

This is used to build a new tree of approximation constants converging towards the value (1/1 + A), as in an example already known from Hightower. The related equation is:

We then build a new diophantine equation:

We show how it is linked to a sequence of approximation constants converging towards (1/3) by lower values.

1. Introduction and Notations

In a former article [ll], the author has produced a new Markoff theory and the associated rules for the corresponding sequences of integers.

The object of the present article is to show how those rules allow to produce many trees of approximation constants, generalizing the constructions of the classical Markoff theory ([8], [I], [3]). In application, an example of a sequence of approximation constants converging towards (113) by lower values is given. In the classical Markoff theory such a sequence is identified, but converging towards the same limit by upper values.

We use the same formalism as coined in [9] and refined in [ll]. We recall it briefly here. Let a finite sequence of integers:

1991 Mathematics Subject Classzfication. Primary 11504, 11506; Secondary 11D09, l lH70, l lK55.

1999 American hlathernatical Soriety

298 SERGE PERRINE

Where: V i = O , . . . , n , a , ~ N*

We associate to S the matrix Ms defined by:

Its determinant is:

And we do the convention that Ma, is the unity matrix. Every sequence S gives its mirror sequence S* = (a,, . . . , a l , ao) and the cor-

responding matrix which is Ms., the transposed one of Ms. To any sequence S and any integer a we associate the algebraic number of

degree 2 whose development in continued fraction can be written, with a periodic part:

Its conjugate is easily computed:

And these two numbers give birth to the binary quadratic form:

If we want to consider only quadratic forms with integers coefficients, we can look at the deduced form:

It is easy to compute Ba(S) with the two following numbers: The antidromic gap:

The discriminant:

So we have:

The approximation constant of Q,(S) which is also the Markoff constant of the forms Fs or fo is the number:

(11) C(FO) = c ( f~ ) = c ( o a ( s ) ) = m a ( s ) l Ja,O Where m,(S) is the arithmetical minimum of the form fe, that is to say the

least value of 1 fo (x, y ) / when x and y are integers not simultaneously equal to 0. Of course, we have:

TREES O F APPROXIMATION CONSTANTS 299

When the situation is simple, as for example in the case of the classical Markoff theory, we have only:

In the rest of this article, we suppose that the sequence S* can be written with two other sequences of integers XI and Xz , and an integer b E N* in the following form:

S* = (XI, 6, X2)

Such a decomposition is also written, when we want t,o identify the structure of S*:

With:

b = a,

XI = (an, . . . ,%+I)

X2 = (a,-1, . . . , a")

We can then consider the two matrices:

(15) %, = [y: f 2 where = det(Mxl) = (- 1)"-' I mz m2 - k.2

(16) Mx2 = I where ~2 = det (Mx,) = (-1)" k.21 k.21 - 12

Writting Ms. with those two matrices, we find many relations among which the following ones:

With the three matrices introduced before, we have a lot of developments in continued fractions, among which:

(24) ml/kl = [a,, . . . , a,+l] = [XI] partial quotient of m/K1

(25) ml l ( m ~ - k . 1 ~ ) = [ a , + ~ , . . . , an] = [XTI

(26) m2/(m2 - k2) = [ao, . . . , a,-I] = [X;] partial quotient of m/(m - K2)

(27) m2/k21 = [a,-, , . . . , a01 = [X2]

300 SERGE PERRINE

Those relations show that the numbers m, ml , rn2 are positive terms. We have also:

(28) a + r + 1 = a + (l/[S]) + (l/[S*]) > 0

As for the form Fs, it gives the following relations:

(29) Fe(K1, m) = -E

(30) Fe(K2 - (a + l ) m , m) = -E

And if we introduce now the number A given by:

(31) Fo(h ,m1) = A

We easily find with the expression (6) of Fe that we have:

This can be compared to the other expression coming from the development of the matrix Ms. :

Considering now the new quadratic form cP0 built with z = rnx - K1y

(34) cPs(z, y) = m"e(x, y) = z2 + ((a + 1)m + K I - K2)zy - &y2

We can compute the number ae(-&lrnz, ml) . With t,he relat,ions (17) to (21) already met. we find the following equality called (Ad:), and generalizing the classical diophantine relation of Markoff ([I], [3] or [8]):

In [ll], we mainly studied the case when E I = ~ 2 . In fact, replacing the sequence S by SaS, we see that it is only necessary to look at this case, given also by E = -1. Such a transformation does not change the numbers O,(S) and O,(S). The form FH(x, y) also does not change. On the contrary. the value m = rn(S) is replaced by:

And the discriminant A,(S) is replaced by:

(37) A, (SaS) = A, (S) [(a + l ) m + K I - K2I2

In such a general case, the equation (M:) takes a simplified expression called in the rest of this article h I E 1 (a, S,u), where we write 6 = K1 - K2, and A =

- ( (m + u) l m ) :

It can be noticed, with remarks made before, that when u > 0, we must have:

(39) E l = E2 = 1

So we have then to consider the diophantine equation M+(a , 6, u):

With S = 0 and a = 3, we find the cases created in [9]. But as we possibly changed the value m, replaced by m(SaS), the values ml and m2 are possibly not

TREES OF APPROXIMATION CONSTANTS 301

the same as in the former equation (Ad:). There is no difficulty to understand the link between those new values and the old ones.

2. The Already Known New Markoff Theory

The article [ll] deals only with the diophantine equation M f '(3,0, 1):

We showed that the solutions are organized in a tree similar to the one existing for the classical Markoff theory [I]. So the fundamental solution (1,1,1) gives all the other solutions with the three following transformations:

(42) X : (m, ml , m2) - (4m,m2 - m - 1, m1, mz)

(43) Y : (m, m1.m2) - (m, 4mmz - m1, m2)

(44) Z : (m, m1, m2) H (m, ml , 4mm1 - m2)

Except the initial solutions (1,1,1) and (2,1, I ) , we can consider the Cohn triples (m, m l , mz), and organize them in another tree explicitly built in [ l l ] . These triples are defined (see [2]) by the condition:

To each Cohn triple, we can associate a sequence S* having the expression (14). Here, we have a = b = 3. We also showed that we can write:

Where T is a palindromic sequence (or mirror sequence), which means:

And the sequence (a(X;)) defined with X; as now described. For any sequence long enough of integers:

(48) X = (02, of-1 7 . . . , o o )

We define:

( a x ) = ( I , % - ~ , Q ? I , . . . ,no) (aa- l+1 ,a , -2 , . . . , a ~ ) i f o , = 1

And combining those two former operators with palindromy:

It is necessary, for any calculus to be possible, to introduce some conventions, understood with the associated matrices, such as:

Every sequence S* resulting from the equation M+(3,0 ,1) can then be built with only the two sequences X2 = 0 and T = 0. The rules t,o apply for building

302 SERGE PERRINE

such a tree of sequences associated to the tree of Cohn triples can be put in a new form. Beginning with:

We have on the left:

This can be expressed by two simplified relations:

On the right, the things are more complicated, and we had to separate two cases:

1) We have for two consecutive steps on the right:

This can be expressed by the two following relations:

2 ) We have for one step on the right consecutive to a step on the left:

This can be expressed also by two relations:

It does not seem possible to give such pretty formulas for only one step on the right. But in fact, it is not necessary, because the formulas given before are sufficient to build the entire tree of sequences. We could think that a more convenient tree could be presented for the same triples and corresponding sequences, but the author of the present article has not yet found it. Anyway, for the case studied here, the top of the tree of sequences can be described by Table I.

The same general relations (51) to (60) can be used to present the classical Markoff theory, which is linked to the diophantine equation M+(2,0 ,0) . In this case, we take a = b = 2. All what we have described can be immediately transposed. At the beginning of the tree, we use also X z = T = 8. And the formulas for going right or left are the same as those given up.

With such a method, the Cohn triple are again organized in a tree, but this one is very different from the known ones ( see [I] or 131 ). The construction that we have made is very peculiar. In fact, in order to build all the sequences of the


TABLE I. Sequences linked to the equation M+(3,0,1)

Tree of Tree of solutions sequences Cohn triple Sequence X2 Sequence T

- XY XYX XY Z XYXY XYXZ XY ZX XYZY XYXYX XYXYZ XYXZX XYXZY XYZXY XYZXZ XYZYX XYZYZ

0 D G DD GD DG GG DDD GDD DGD GGD GDG DDG DGG GGG

classical Markoff theory, we need only the following Table I1 of what happens at the top of our tree:

TABLE 11. Sequence linked to the equation M+(2,0,0)

Tree of Tree of solutions sequences Cohn triple Sequence X2 Sequence T

3. Example With Constants Converging Towards (1/(1 + &))

Now we use the same relations (51) to (60) as before, but keeping a = b, and without changing the expression (16) associated to Mx,. We suppose that T does not have anymore the property to be palindromic. So we associate to this sequence a matrix:

We can then compute Mx, and M p . This gives all the numbers m, K1, K2,1,

etc.. .

304 SERGE PERRINE

In order to produce a readable example, we simplify here the calculus, supposing that we have X2 = 0. We then find very easily the following relations:

So:

In this case, the fact for T to be palindromic ( T = T*) is equivalent to the fact that the antidromic gap of S* is equal to zero. Another equivalent condition is 6 = 0. This is no more true in our general case for T, where we find many relations:

The expressions found for m, ml and m2 allow to verify a general diophantine equation of the form (38). In fact, we can compute:

This gives the possibility to have an expression of A. Substituing in the determinant of My. we find a new diophantine equation (EQ') which can be written, with (67), a = band y = 7,:

We now see that when a , 6, and u are given, any solution (p, A) of this equation gives At7,. In fact usually, such a solution gives birth to an infinity of solutions, because the equation is quadratic, and we can use a group of hyperbolic rotations leaving the equation unchanged. We then find an infinity of solutions of the diophantine equation M+(a , 6, u) .

Working on an example, we see now how all this works. Let us choose here ; t

T = (1,2). We find S* = (1,1,1,2,2, 0 ), with a = b = 2, and X2 = 0. In this case, it is easily seen that 6 = 1 and u = 2. So we consider the diophantine equation M+(2,1,2):

The sequence S* identified before gives the following triple of solutions:

TREES OF APPROXIMATION CONSTANTS 305

With such a solution, we find a infinity of other solutions using the three transformations:

(77) X : (m, rnl, m2) - (3m1m2 - m - 2, ml , mz)

(78) Y : (m, m1, mz) - (m, 3mm2 - ml + m:!, m2)

(79) : (m, m1, m2) - (m, m1, 3mm1+ ml - m2)

Applying for example X Y we find the new triple of solutions:

In fact, looking at the calculus made before for the transformation DD, we see that this is precisely the transformation XY. So we find with (56) and (57) that the new X2 is again 0, and that the new T is (1,1,1,2,1,1). The new S*

C

corresponding to the last triple is so (1,1,1,1,1,2,1,1,2, 0 ). We can verify it directly by the expressions (17) to (27), with K1 = 80, K2 = 79, kl = 31, k2 = 1.

More generaly, the relations (55) to (57) leads to consider the transformation of the first part of S* given by:

It is easy to see that it gives an infinity of sequences S*. All of them are associated to our new equation (76).

With XI we can easily find another quadratic diophantine equation equivalent to ( E Q ~ ) . For that, we use the relation (32) under the new form:

The values of u, k2, m:!, are known, giving:

Replacing in the equation (76) and simplifying, we find the equation (EQ):

We verify that (ml , kl) = (50,31) is a solution, and that we can find again the equation ( E Q ~ ) writting Mx, with MT. With the relation (20) we can compute K2 :

With the definition of S = 1:

It is now clear why the relation (80) seen before between X1 and ~f gives a lot of solutions of (EQ) and (EQT).

With the Dickson lemma (see [5] vol2, pp. 408-409), we can then compute the approximation constants of the algebraic numbers associated to the sequences S* that we consider. We will not do it here for all the tree derived from the equation M+(2,1,2) , but it is possible to do it. For rapidity, we will do it only for the sequences built with (80).

306 SERGE PERRINE

The algebraic number to look at is of the form:

Its constant of approximation is given by the greatest of the three following numbers (see [3]):

We easily see that:

The comparison between S1 and 63 is a little more difficult, but not too much. We see that 61, and 63b are very near, with < Ssb, though 63, and 6lb are quite far from each other, with 63, < Sib. SO S1 is the greatest of the three considered numbers, giving the approximation constant:

Using the values (4) of e2(S) and ( 5 ) of 02(S), this constant can be computed. Also:

Those two expressions can be given with only ml and kl, which are linked by (83). It is easy, with (88), to see that the limit of the constant when n increases is (1 / (1+ A)). The limit of 6;' can also be computed and is (113). In fact, we can write:

Where we have in comparison with (12):

In the relation (31) we could put:

This would give a new diophantine equation similar to (38) with a tree of solutions (see [6]).

The comparison with a result of Hightower [7] is also interesting, because with our new example, we can understand the gaps discovered by this author.


4. Some Answers to Old Questions

Now, we look again at the results given in [9]. We gave there a table of all the sequences appearing in the so called (3,0, -1) theory. We saw that different values were possible for u, and computed the different sequences S* associated to those values of u. With a calculus similar to the one made in the former paragraph, we now understand why all the sequences appearing can be built with a sequence X2 and a palindromic sequence T = T*. This result comes from the fact that we have supposed that K1 = K2 for the sequence S*. For example, with m < 300, we find:

TABLE 111. Analysis of the sequences of the (3,0, -1) theory

We use the computations made before, with Mx, being any matrix built with X2 sequence of integers, MT given the same way but with T palindromic, and a = b. We find:

Now the computation shows really that:

It is also easy to compute m, 1, and u:

Practically, when X2 and T have an even number of terms, we have ~1 = € 2 = 1. We find only equations of the type M f (a, 0, u) with u the last form given. And we have given a way to build an infinity of values of u for which such an equation have at least a tree of solutions. It is known that not all the values of u are possible (see [4] and [lo]).

308 SERGE PERRINE

With the calculus that we just made, we can choose X 2 = 0, and T =

So we find with the Fibonacci numbers:

With a = 3, we see now that:

We have then the demonstration of a remark made in [9] that all the diophantine equations M+(3,0 , F2n) have at least a tree of solutions, and we know how to build one.

5. Constants Converging towards (1/3) by Lower Values

In order to build such a sequence of constants, the idea is to be out of the classical Markoff theory, but to keep its main features.

We introduce the sequence:

The Dickson lemma gives the possibility to compute the corresponding approximation constant. With the same method as used in the former paragraph, we have to consider the greatest number among the four ones:

We see that d3 = S2 and b4 = 6i. It is also easy to verify, with the same notations as before, that we have:

So, the approximation constant is in this case:

To be coherent with such a value, the sequence now considered is:

When i is increasing, the value of C converges towards (113). And because the subsequences of 1 appearing in S* have an odd length, we are not in the classical Markoff theory. So we have:

We have built the promised sequence of approximation constants converging towards (113) by lower values.


In the case where j = i + 1, we can write:

We find, with such conditions:

And XI and X 2 give the sequence S* with a relation like (14). Our formalism now creates a tree of sequences applying the rules (52) to (61). Because T is palindromic, we have always K1 = K2. The corresponding diophantine equation have the form M+(2,0, u ) . With the methods already known, we can compute u . Thanks to Fibonacci numbers, we find always the same value:

We can also compute the different values of (m, ml , m2) found before and giving a triple of solutions of our diophantine equation M+(2,0, -2):

So we have established:

THEOREM 5.1. Let us consider the diophantine equation M+(2,0 , -2)

Then the triple of numbers:

is a triple of solutions of this equation. Moreover, for each such triple, we can associate a algebraic number of degree

2, and the corresponding form, whose constants are less than (113). W h e n i increases, those constants converge by lower values towards (113) which

is their accumulation point.

REMARK 5.1. With the given triple of numbers in the equation M+ (2,0, - 2 ) , a simplification by m is possible. It leads to:

Or: 2

F2t+4 - F2i+~fiz+3 = -1 This equality is an evidence with:

We have so a direct verification of the main part of our theorem. The rest can also be verified with similar methods to those used in [ l l ] .

310 SERGE PERRINE

References

[I] J.W.S. Cassels, A n introduction to dzophantzne approximation, Cambridge tracts in Math. Physics 45 (1957).

[2] H. Cohn, Representatzon of Markoff's binary quadratzc forms by geodesics on a perforated torus, Acta Arithmetica XVIII (1971), 123-136.

[3] T.W. Cusick and M.E. Flahive, The Markoff and Lagrange Spectra, Mathematical surveys and monographs, 30, A.M.S., 1989.

[4] T.W. Cusick, O n Per-rine's generalized Markoff equation, Aequationes Mathematicae 3 (1993), 203-211.

[5] L.E. Dickson, History of the Theory of Numbers, Chelsea reprints, New York, 1992. [6] El Khuti El'Huti (M.Kh.), Cubic surfaces of Markoff type, Mat. Sb. 93 (1974), 331-336. [7] C.J . Hightower, The mznzma of indefinzte binary quadratic forms, J . Number Theory 2 (1970),

364-378. [8] A.A. Markoff, Sur les formes quadratzques inde'finies, Math. Ann. 6; 17 (1879; 1880), 381-406;

379 399. [9] S. Perrine, SUT une ge'ne'ralisation de la the'orze de Markoff, J. Number Theory 37(2) (1991),

211-230. [lo] S. Perrine, Sur des equations diophantzennes ge'ne'ralisant celle de Markoff, Annales de la

Facult4 des Sciences de Toulouse VI(1) (1997), 127-141. [Ill S. Perrine, Un arbre de constantes d'approximation analogue a celui de l'e'quation diophanti-

enne de Markoff, submitted.

5 RITE DL' BON PASTEUR, 57070 METZ, FRANCE E-mail address: Serge. PerrineQwanadoo . f r


Continued Fractions and Schrodinger Evolution

Igor Rodnianski

To Jerry Lange

ABSTRACT. We study the spatial regularity in Besov spaces of the fundamental solution E( t , x) of the time-dependent Schrodinger equation on a circle in the presence of a complex-valued potential V. Although the fundamental solution is not smooth, we reveal a fine change of regularity of E(t ,x) a t different times t. For rational t, E(t , x) exhibits the same regularity as at t = 0. For irrational t , the regularity of E(t , x) is better and depends on how well t is approximated by rationals. For badly approximated t (e.g., when t is a quadratic irrational, or, more generally, when t has bounded quotients in its continued fraction expansion), E(t , x) is a " 112-derivative" more regular than E(0, x). For a generic irrational t , E(t , x) is almost " 112-derivative" more regular. However, the better t is approximated by rationals, the lower is the regularity of E(t, I ) . We describe different thin classes of irrationals which prescribe their particular regularity to the fundamental solution. We establish the results by representing the fundamental solution E( t x) by a sum of the fundamental solution for the free Schrodinger equation (V(x) 0) and a smoother function.

1. Introduction. Evolution of the quantum mechanical particle can be described by the time-dependent Schrodinger equation with quantum Hamiltonian H

In the case of a particle of mass m on the circle T = R/Z in the electric field of the potential V(x), the Hamiltonian H can be represented as follows:

1 d" H = + V ( x), x E T.

2m dx2

1991 Mathematics Subject Classif?cation. Primary 81Q05, llL03, 35D10; Secondary llL07, 46335.

Key words and phrases. Schrodinger evolution, Besov spaces, continued fractions. Research was partially supported by NSF grant DMS-9623520


311

312 IGOR RODNIANSKI

The Schrodinger equation (1.1) takes the form:

For convenience, we replace equation (1.2) by the dimensionless equation

Initial value boundary problem (1.3) can be thought of as the time-dependent Schrodinger equation on the interval [O,1] with periodic boundary conditions.

In the case of a free particle ( Ho = -& & ) the solution of equation (1.3) can be written explicitly as:

where are the Fourier coefficients in the expansion

The kernel of the propagator e P t H " i s the solution of equation (1.3) with V(x) - 0 and the initial data

It can be represented by a formal series

Note that when t is in the upper half-plane, then Eo(t, x , y) is (essentially) Jacobi's

n

The solution operator ePtH0 is a unitary operator in the scale of Sobolev spaces Hs := H"(T). Recall that function f = xnEZ fne2xznx E H S if and only if

Clearly, for each y, E( t , . , y) E n Hpa-' for all t 2 0 and does not belong to a c > O

better Sobolev space . This shows that the series in (1.5) does not define a smooth function.

The question of regularity for E ( t , x , y) fits in the general framework of the study of the smoothness of the fundamental solution for a Schrijdinger evolution.

The Schrodinger equation on Rn has been extensively studied. In that case, in general, one can expect that the fundamental solution becomes infinitely smooth for positive times. In fact, this has been proved for a wide class of Hamiltonians H with potentials V growing slower than quadratically at infinity, see [Z,1983],

[KR,1996], [Y,1996]. Moreover, a similar result has been proved for the more general class of equations with variable coefficients. The smoothing of the fundamental solution occurs provided bicharacteristics of the corresponding classical Hamiltonian are not trapped, see [CKS,1995], [KS,1996]. The situation of the Schrodinger equation on more general than Rn non-compact manifolds has been addressed in [W,1998].

In the case when the underlying space is compact, one can no longer expect the fundamental solution to become smoother for positive times. This is supported by the results of K. Yajima [Y,1996], who showed that in the case of the Schrodinger equation on an interval with Dirichlet boundary conditions, the fundamental solution is nowhere smooth, and by the work of S. Doi [Do,1996].

In the paper [KR,1998] we proposed to address the question of regularity of the fundamental solution Eo(t , x , y) for the Schrodinger evolution on a circle from the point of view of Besov spaces B;,, rather than Sobolev spaces. We discovered that although the fundamental solution is not smooth, its regularity is changing with time. Regularity at the moment in time t is related to arithmetic properties of t. For all rational times Eo( t , ., , y) E B,:,, when for a generic irrational time

t , Eo(t , ., y) is almost " ;-derivative better". It belongs to the space n B;!;'. In r>O

general, the worse t is approximated by the rationals, the better the regularity of the fundamental solution at time t.

The Besov spaces' approach requires investigation of the partial sums (incomplete theta-functions)

For every N and fixed t and x, (1.6) determines a point on the complex plane. Connecting the points with consecutive N by straight lines we obtain interesting geometric figures, curlicues in the terminology of M. Berry and J. Goldberg [BG,1988]. Their pattern depends upon the arithmetic properties of the parameter t , in particular, on the closeness to the set of rational numbers. Similar sums appear in the diffraction experiment with planar waves incident to a periodic grating with N narrow slits. The intensity of light on the screen behind the grating is determined by the square of the absolute value of the sum in (1.6), where the role of parameter t is played by the distance to the screen. Therefore, the image on the screen depends upon arithmetic properties of the distance. M. Berry and J. Goldberg suggested to call this experiment the arithmetic microscope [BG,1988]. The studies of the geometric patterns generated by theta-sums are also contained in [DM1981], [D,1985], [BH,1987], [CK,1987].

In this paper we address the question of regularity of the fundamental solution for the Schrodinger evolution on a circle in the case of a non-zero complex-valued potential V(x). We show that the picture observed in the case of a free particle on a circle is completely stable for a wide class of potentials. On the compact manifold, a smooth bounded potential V(x) constitutes a weak perturbation of the Hamiltonian Ho. Therefore, one might expect that qualitatively the effects observed in the case of the free particle will remain the same for this class of potentials. The surprising element is that the behavior of the fundamental solution is exactly the same as in the case of the Hamiltonian Ho. This stability suggests

314 IGOR RODNIANSKI

that should any physical experiment relating the smoothness properties of the wave function at time t and arithmetic properties of t be carried out, one could neglect the contribution of the electric field.

The effect can be best demonstrated in the special case of a real-valued potential V. In that case we have the following representation for the fundamental solution:

where An are the eigenvalues of the operator - & & + V(z), and 4, are its eigen- functions. Treating the potential V as a perturbation, we can rewrite E ( t , x, y) in the form Eo(t, x, y) + error and show that the error term defines a function smoother than Eo (t, x, y) .

We should mention that K. Oskolkov investigated the pointwise behavior of the solutions of periodic Schrodinger equations, see [0,1992] and references therein. In particular, he investigated the series

and proved that it defines a continuous function in x for every irrational t . It is discontinuous at all rational t where the sum ~ r = ~ ~ " " ~ ~ ~ + ~ ~ ) # 0.

ACKNOWLEDGMENTS. The author thanks Lev Kapitanski for valuable discussions.

2. Statement of results. We start with the definitions of the Besov spaces. For reference on Besov spaces,

see [BL,1976], [T,1983]. Let x be a C(y function on R+ with the following properties:

and m

Define the functions

With each of these functions we associate an operator K j , which maps a distribution f (x) = 1, f , e2"imx to a (finite) exponential sum,

We define the Besov spaces B;,, on T for the following values of parameters s,

p, and q: - m < s < + m , 1 5 p, q < +m. By definition, the space B;,, is

CONTINUED FRACTIONS AND SCHRODINGER EVOLUTION 315

composed of all distributions f ( x ) = Em f , e2"imx such that the following norm is finite:

In this paper we use the spaces B& most frequently. Therefore, we make a simpler notation for them: B&.

We shall say that a distribution f belongs to B& sharp if f E B& and f ?? ut>o B Z .

The fundamental solution E(t , x , y) of the Schrodinger equation on the circle T with the potential V is a solution of the following initial value problem:

For simplicity, we shall consider the function

Regularity properties of E ( t , .) depend upon the behavior of the continued fraction expansion of t . In the case of a rational t = :, the continued fraction is finite and has the form

P 1 t = - = [ a o , a l , . . . , a,] = a ( )+ 4 1

a1 + - 1 - an

Note, that the expansion is not unique: we also have = [ao, a l , . . . , an- 1, a,- 1,1], if a, # 1, and 2 = [a0, a l , . . . , an-1 + 11, if an = 1.

4 For an irrational t the continued fraction expansion

is unique. Quotients a,, can be found from the recurrent relations

1 1 a k + ~ = [ - I , tlci-1 = - - ak+l,

t k t k

and the initial conditions

ao = [t] , to = t - a0 .

As usual, [r] denotes the largest integer not exceeding r .

316 IGOR RODNIANSKI

The finite parts [ao, a l , a2, . . . , a,], n = 1, 2 , . . . , of our infinite continued fraction, sum up to the rational numbers p,/q, - partial convergents of t . The numerators and denominators of the convergents can be found using the relations

and the initial conditions

Of course, p,/q, + t . Also, we have

For the basic theory of continued fractions, see [K,1964] and [S,1980]. We state our main result.

THEOREM I . Let E ( t , x) be a solution of equation (2.4) with the initial condition

Assume that a complex-valued potential V belongs to the space U H $ + € . t>O

Denote

Let E o ( t , x) be the corresponding solution of a free equation (V(x) - 0). Then, the fundamental solution E ( t , x) can be represented i n the following form:

where F ( t , .) E n B;ct for any fixed t . t>O

REMARK 2.1. The definition of Besov spaces (2.3) implies that for all t func- 1 1

t ion E o ( t , .) belongs to the space B2:z sharp. Because of the embedding B G L - 1

B i z , we can conclude that, for all t , E o ( t , .) does not belong to a space better

than B,'. Therefore, Theorem I implies that E o ( t , .) completely determines the regularity of E ( t , .).

We can simply reiterate the known regularity properties of Eo(t , .). We use the results obtained in [KR,1998]. Note that E o ( t , x) is E ( t , x) in the notations of [KR,1998]. In the case of a rational time we have the following theorem (compare [KR,1998, Theorem I]).

THEOREM 11. Let t be a rational number, and t = - its simple fraction representation. Let E, k = 1, . . . , n - 1, be partial convergents to t determined by a finite continued fraction [ao, a l l . . . , an ] with a n odd number of quotients (i.e., n is even). Then,

I ) (formula)

where E = ~ . q ( m o d 2 ) , T = P , - I . ~ , - I ( ~ o ~ ~ ) ,

and x o ( t ) is a n eighth root of 1; 2) (regularity)

E ( t , .) E B & ~ sharp.

The next theorem establishes regularity of E ( t , .) for a generic irrational t , and shows the relation between the smoothness of the fundamental solution at the moment in time t and how well irrational t is approximated by its partial convergents (compare [KR,1998, Theorems I1 and 1111).

qn We define the following classes of irrational numbers. For a > 0, denote by

J>, the set of all irrational t such that there exists a constant ct such that for al l partial convergents 2 with sufficiently large n

Denote by J<, the set of all irrational t such that there exists a constant ct such that for an infinite number of n

Finally, denote Ju = Js, n J?,. Note that in [KR,1998] we introduced the following classes of irrational numbers: I>,, I>, and 2,. They were characterized by the rate of growth of the denominators ;f the partial convergents. Namely, t E Z>, if for an infinite number n there exists a constant ct such that q,+~ >_ ctqA+"; t E I<, if for all sufficiently large n there exists a constant ct such that qn+l < ctqAf u: and 2, = I>, n I>,. - It is easy to check that

THEOREM 111. ( i ) For almost all irrational t , E ( t , .) E n B,""'

t > O

(ii) If t is an irrational number with bounded quotients, i.e., there is a constant C > 0 such that a,, < C , for all n, then E( t , . ) E B,"~ sharp. (iii) If t E J?, , then

(iv) If t E J5,, then

318 IGOR RODNIANSKI

(v) I f t E ,Xu , then 1+c --

E ( t , .) E B,~+~ .

REMARK 2.2. The celebrated Roth's theorem implies that any algebraic number

t E uT0,X2C. Therefore, for any algebraic t , E ( t .) E n B&'-'. t>O

REMARK 2.3. From the results of [KR,1998] it follows that Theorems 1-111 also hold for the so-called rough Besov spaces [Bl&. Their definition differs from that of B&,,, practically, only in the choice of the cut-off function in (2.1): instead of a smooth X , we take a step-function [ X I - the characteristic function of the interval [2-', 21.

3. Proof of Theorem I. Consider the Schrijdinger equation on a circle

with the initial condition

Under the assumption of Theorem I on the potential V, there exists a unique solution of problem (3.1), E E C ( R ; n H-~- ' (T)) . This can be demonstrated for

t > O example, by the Galerkin method. Denote

Rewrite equation (3.1) in the following form

Introduce the operator

Duhamel formula transforms (3.3) into the integral equation

It is easy to see that e i A t ~ O is a solution of equation (3.1) with the constant potential 7rVo and the initial data e2"inx . Therefore, as in the case of a free equation, it can be represented by the series


where Eo( t , x ) is a solution of equation (3.1) with the potential equal to zero. The right-hand side of (3.6) coincides with the first term of identity (2.8) in Theorem I. Therefore, we need to show that for all t , function

belongs to the space n BGt. c > o

Denote by Pk the operator of projection on the function e2"'" - the kth d2 eigenfunction of the operator -. Clearly.

Apply operator Pk to both sides of (3.5). Using (3.8) and (3.9) we obtain (3.10)

t

Obviously,

Iterating equation (3.10) we have

Therefore.

320 IGOR RODNIANSKI

Integration by parts in (3.11) yields (3.12)

1 J - + - - e-TL("+~~l)( t-~) )P~VP[V(I - P/)E(T, .) d r . 7ri(k2 - 12)

0

We have the following equality

Ph VP, =

where V,, = ~(r)e- '"" '" dx is the n th - For k # 0 let us estimate

Vk-lP1 r

Fourier coefficient of the function V(x).

Coefficients V,, are uniformly bounded by 1 IVl l L l = IV(x) ldx. Also. as it follows T

from (3.lb). El' = 1 for all 1 E Z. Therefore,

IVrite 1 in the form 1 = rnk + n. We have

Sinlple computation shows that the first two terms can be bound by c w. The

last term on the right-hand side of (3.14) can be estimated by

1 1 c

Therefore,

We also need to estimate

Using Holder inequality we obtain

for some E > 0. The fundamental solution E ( T , .) belongs to the space n H - $ - ' . e>u

Also, by the assumption of the theorem, V E U H++'. Therefore, with the help t>O

of the inequality

and the estimate IVk-ll < liVlILl we have

Repeating the argument used to derive estimate (3.15) we obtain

Combining estimates (3.15) and (3.19) we conclude that for any E > 0 and t E R there exists a constant c ( t , E ) such that

Therefore, F ( t , .) E n H-'. It follows that t > O

We can replace equality (3.12) by the the following identity:

322 IGOR RODNIANSKI

The bound for the first term on t,he right-hand side of (3.22) has been found in (3.15). For the second term. we have similar to (3.16)

Following the estimates leading to (3.18) but taking into account, (3.21) we have

The last term can be integrated explicitly:

and

The right-hand side of (3.24a) can be estimated by

where we made use of the following:

Also.


Combining (XIS), (3.23) and (3.25) we obtain the refined estimate on 4 F ( t , .) : ln lkl

P k F ( t , ) / < c ( t , f ) w . 'dk E Z\ (01,

which holds for all c > 0 and some constant c( t . F ) . It follows that F ( t , .) E n H 4-' . Embedding n H 3 ~ ' - n B;' implies the conclusion of the t heorern.

c > O t > O ' > O n

References

[Do,1996]

[HB,1980]

[KR,1996]

[KR, 19981

J. Bergh and J . Lofstrom, Interpolation spaces, an zntroductzon, Springer, Berlin, 1976. hl.V. Berry and 3. Goldberg, Renormalzzatzon of curlzcues, Nonlinearity 1 (1988), 1-26. E. A. Coutsias and N. D. Kazarinoff, Disorder, renorma~zzabi~zty, theta functzons and Cornu spirals, Physica D 26 (1987), 295-310. W. Craig, T . Kappeler and W. Strauss, Microlocal dzsperszve smoothing for the Schrodinger equation, Comm. Pure Appl. Math. 48 (1995); 769-860. F. M. Dekking and M. Mend& France, Un,zform dzstrzbution modulo one: a geometric view point, J. Reine Angew. Math. 329 (1981), 143-153. 3.-M. Deshouillers, Geometric aspect of Weyl sums, Elementary and analytic theory of numbers, Banach Center publications, volume 17, PWN-Polish Scientific Publish- ers, Warsaw, 1985, pp. 75-82. S. Doi, Smoothzng effects of Schrodinger evolution groups on Rzeman,nzan manzfolds, Duke Math. J . 82 (1996), no. 3, 679-706. J . H. Hannay and M. V. Berry, Quantzzation of h e a r maps on a torus Fresnel dzffraction by a perzodzc gratzng, Physica D 1 (1980); 267-290. L. Kapitanski and I. Rodnianski, Regulated smoothzng for Schrodznger evolutzorz, International Math. Research Notices 2 (1996), 41-54. L. Kapitanski and I. Rodnianski, Does a quantum particle know the tzme?. to appear in Proceedings of the workshop on Emerging Applications of Nunlher Theory. ILIA Volumes in Mathematics and its Applications, 1998. L. Kapitanski and Yu. Safarov, Disperszue smoothzng for Schriidznger equatzons, Math. Res. Letters 2 (1996), 41-54. A. Ya. Khinchin, Contznued fractzons, Translated from the 3rd Russian edition of 1961, The University of Chicago Press, 1964. P. I. Lizorkin, On bases and multipliers for the spaces BF,o(II), Trudy Mat. Inst. Steklov 143 (1977), 88-104. I(. I. Oskolkov. A class of I. M. I/zn,ogradov's series and its applications zn harmonic analysis, Springer Series in Computational Mathematics, 19, Progress in Approxi- mation Theory, an International perspective (A. A. Gonchar and E. B. Staff, eds.), Springer-Verlag, New York, 1992. Schmidt, Diophantzne approximatzon, Lecture Notes in Mathematics, 785, Springer- Vcrlag, Berlin Heidelberg New York, 1980. H. Triebel, Theory of functzon spaces, Birkhauser, Basel, 1983. J . Wunsch, Mzcrolocal analysis of the time-dependent Schrodznger operator, Thesis, Harvard University (1998). K. Yajima, Smoothness and non-smoothness of the fundamental solutzon of tzme dependent Schrodinger equations, Comm. Math. Phys. (1996) (to appear). S. Zelditch, Reconstructzon of singu~arities for solutaons of Schrodinger's equatzon, Comm. Math. Phys. 90 (1983), 1-26.

DEPARTMENT OF MATHEMATICS. KANSAS STATE I.NIVCRSII 'Y, ILIAN~I,VI'TAU. K S (Xi506

E-mail address: ir0dmmath.k~~. edu

Contemporary Matherrlatic-s Volume 236. 1999

Multiple orthogonal polynomials, irrationality and transcendence

Walter Van Assche

Dedzcated to Jerry Lunge at the occaszon of hzs 70th bzrthday

A B S T R A C ~ We explain the notion of multiple orthogonal polynomials (polyorthogonal polynomials), which is closely related to Hermite-Pad6 rational approximation of a system of hlarkov functions, and give some explicit examples. As an application we show how multiple orthogonal polynomials can be used to give a constructive proof of irrationality of certain real numbers and also of transcendence of real numbers. Historically Hermite-Padk approximation was introduced by Hermite to prove the transcendence of e.

1. Orthogonal polynomials

The notion of orthogonal polynomials is an old one. going back to the previous century (Chebyshev, Stieltjes). A very good source of information is Szego's book [Sz] and a more recent exposition can be found in [VAl]. hlonic ortilogonal polynomials for a positive Bore1 measure p on the real line are polynomials P,, (n = 0 ,1 , . . . ) of degree n and leading coefficient one such that

These n orthogonality conditions given n linear equations for the n unknown coefficients ah ,, ( k = 0,1 , . . . , n - 1) of the rnonic polynonlial p , , (~) = x:=,, a k ,,z I, (where a,, , = 1). This system of n equations for n unknowns always has a unique solution since the matrix with entries xl'J d p ( x ) (0 5 z. J < n - 1) (this is known as the Gram matrix) is positive definite and hence nonsingular. It is well known that such polynomials satisfy a three-term recurrence relation

1991 Mathematzcs Subject Classzficatzon. Primary 42C05. 33C45; Secondary 11372, 11382. Research Director of the Belgian National Fund for Scientific Research (FWO). This research

is supported by FWO research project G.0278.97 and INTAS 93-219ext.

@ 1999 Arnr r l r an hlathematical Society

326 WALTER VAN ASSCHE

with f i = 1 and PP1 = 0. Often it is more convenient to consider the orthonormal polynomials p,, (x) = y,, P, (x) for which

so t>hat yic = (I P:(x) d p ( ~ ) ) - " ~ > 0. The recurrence relation for these orthonor- nlal polynomials is given by

where

A very important result in the theory of orthogonal polynornials, known as Favard's theorem. say that a system of polynomials P, (n = 0,1 ,2 , . . . ) satisfying a three- term recurrence relation of the form (1.2) with a: > 0 for all n > 0 is always a system of monic orthogonal polynomials for some positive Bore1 measure p on the real line. Hence the orthogonality (1.1) and the recurrence relation (1.2) are two equivalent ways to describe monic orthogonal polynomials.

Starting from the recurrence relation (1.5) we can construct the Jacobi matrix

The recurrence relation (1.5) can then be written as

from which we conclude that every zero xJ ,, of pn is an eigenvalue of J,, with eigen-

vector (po(x, ,,), pl (J, ,,), . . . , p, -~(x , , , ) ) I . Since all zeros of orthogonal polynomials are simple, it follows that the eigenvalues of J, are the same as the zeros of p,, . One can also consider the infinite matrix J , which acts as an operator Jx : t 2 4 t 2

on an appropriate domain. The symmetric matrix J , has selfadjoint extensions to its maximal domain (sometimes several) and the study of such a selfadjoint operator (especially the spectral theory of such operators) gives useful insight in the corresponding orthogonal polynomials.

Let us mention at this point the best known examples: the very classical orthogonal polynoinials of Jacobi, Laguerre and Hermite These three families of orthogonal polynomials satisfy a second order linear differential equation, their derivatives are again orthogonal polynoniials of the same family (but with different

MULTIPLE ORTHOGONAL POLYNOMIALS 327

parameters) and they can all be obtained from a Rodrigues formula. These properties can be extended to difference operators and q-difference operators and give the classical orthogonal polynomials in the Askey table [KS].

1.1. Jacobi polynomials. The monic Jacobi polynomials P:~") satisfy the orthogonality relations

where a , @ > -1. There are two important differential operators, namely the lowering operator D for which

and the raising operator D,,P = (1 - ~ ) - ~ ( 1 + X ) - ~ D ( ~ - x)O(l+ x)P for which

Both (1.8) and (1.9) can be proved using integration by parts and the orthogonalitmy (1.7). Combining the raising and lowering operators gives the differential equation

where y(x) = P?")(X), and repeated use of the raising operator DaX3 gives the Rodrigues formula

Using Leibnitz' formula we can obtain the explicit formula

(1.12)

The historical way to normalize the Jacobi polynomials is by setting Pi" "'(1) =

(TLLO) SO that p!, .:')(x) = 2 ~ " . ) ~~j"" ' ) (x ) . See Saeg6 [Sa , Chapter IV] for more properties of the Jacobi polynomials.

1.2. Laguerre polynomials. The monic Laguerre polynomials LE satisfy the orthogonality conditions

(1.13) ~ ~ ( x ) x " e - ~ x ~ dx = 0, k = 0, 1 , 2 , . . . , n - 1, I" where cr: > -1. Differentiation gives again the lowering operation

and D, = x-"exDxae-" is the raising operator for which

(1.15) x~~ L; (2) = - L;;: (x).

328 WALTER V A N ASSCHE

Both (1.14) and (1.15) can easily be proved using integration by parts and the orthogonality relation (1.13). Combining the lowering and the raising operator gives the differential equation

where y ( x ) = L ; ( X ) . Using the raising operator repeatedly gives the Rodrigues formula

from which one can obtain an explicit expression using Leibnitz' formula

The historical normalization for the Laguerre polynomials is to take LE(O) = (":(I)

so that L : ( x ) = ( - l ) ' " n ! ~ : ( x ) . See Szeg6 [Sz, Chapter V] for more properties of Laguerre polynomials.

1.3. Hermite polynomials. The monic Hermite polynonlials H,, satisfy the orthogonality relations

Integrat,ion by parts and the orthogonality show that D is a lowering operator

and we have the raising operator

Combining the lowering and the raising operator gives the differential equation

(1.22) y " ( x ) - 2 x y 1 ( x ) = - 2 n y ( x ) ,

where y ( z ) = ~ , ( r ) . Repeated application of the raising operator gives t,he Ro- drigues formula

and by using Faa di Bruno's formula for the nth derivative of a composite function we obtain

Historically the Hermite polynomials are normalized to have leading coefficient 2", hence H,,(z) = 2'"1, ( x ) . For more properties we refer to Szegij [Sz, Chapter 51.


2. Multiple orthogonal polynomials

During the past few years there has been increased interest in multiple orthogonal polynon~ials, particularly in Eastern Europe. See, e.g.. surveys by Aptekarev [AZ], de Bruin [dB] and Chapter 4 in the book of Nikishin and Sorokin [NS]. For multiple orthogonal polynomials we will need multi-indices consisting of r positive integers. for which we use the notation Z = ( n l , nz, . . . , n , ) E N7. where r E N. Furthermore we will use the notation 161 = nl + n2 + . . . + n,. We distribute the orthogonality conditions over r real intervals A l , A 2 , . . . . A, with r different measures pl , p2, . . . , p,. TWO different ways are considered, which give type I and type I1 multiple orthogonal polynomials.

2.1. Type I1 multiple orthogonal polynomials. Type I1 multiple orthogonal polynornials are monic polynomials P,I of degree 161 such that

For r = 1 we get the ordinary monic orthogonal polynomials, but for r > 1 we get multiple orthogonal polynonlials (or polyorthogonal polynornials). All together the orthogonality conditions give 161 linear equations for the 161 unknown coefficients

161 ah 5 of the polynomial PC (x) = EL=,, ah ,-xh , where ali; i; = 1. However, the matrix of coefficients of this system can be singular and we need some extra conditions on the r measures p l , p2, . . . , pr in order that the multiple orthogonal polynomial is unique. If the polynomial Pr?(x) is unique, then we say that n' is a norrnal index and if all indices are normal then we have a complete system. Our interest is of course in systems of r measures for which all multi-indices are normal.

2.2. Type I multiple orthogonal polynomials. For type I rnultiple orthogonal polynomials we want to find a vector of polynomials (A: 1 . A,? 2 , . . . , AZ , ) such that each polynomial A<, is of degree n, - 1 and

Again for r = 1 we find the ordinary orthogonal polynomials. but for r > 1 we find multiple orthogonal polynomials. There are in'l - 1 orthogonality conditions which give 161 - 1 homogenous linear equations for the 161 unknown coefficients of all the polynomials Ar? , (3 = 1 ,2 , . . . , r) . Hence we can determine the vector (A< 1 , Aii 2 , . . . , A< ,) up to a multiplicative factor.

2.3. Operator approach. The t hree-term recurrence relation (1.2) and the Jacobi rnatrix (1.6) have an interesting extension for multiple orthogonal polynomials, which was subsequently exploited by Kalyagin [KZ] to initiate the spectral


analysis of certain nonsymmetric operators. We construct a special sequence s'(n) ( n E N) of multi-indices in W" by writing n = kr + j : with 0 5 j < r and then set

These indices are nearly diagonal and if all these indices are normal, then we have a weakly complete system. Now consider the monic type I1 multiple orthogonal polynomials with these indices, i.e., we put Q,(x) = P,-(,)(x), then one can use all the orthogonality relations (2.1)-(2.3) to show that they satisfy a recurrence relation of the form

with initial conditions Qo = 1 and QPl = QPa = . . . = Q-, = 0. These diagonal polynomials therefore satisfy a linear recurrence relation of order r + 1, generalizing the second order recurrence relation for ordinary orthogonal polynomials ( r = 1). We can then construct a banded Hessenberg matrix

Again it turns out that the eigenvalues of H , coincide with the zeros of the multiple orthogonal polynomial Q, = P,-(,,). The matrix H , is not symmetric, thus there is no a priori reason why the eigenvalues should be real. Nevertheless it turns out that in many cases the zeros are indeed real. Hence the coefficients a, (n + i) (n E W ) , with 0 5 i < r are not arbitrary and have some hidden structure. An interesting open problem is to find this hidden structure and to find out under which conditions on these coefficients the matrix H,, corresponds to n~ultiple orthogonal polynomials.

3. Hermite-Pad6 approximation

3.1. Simultaneous Pad6 approximation. Multiple orthogonal polynomials are closely related to simultaneous Padk approximation, which is nowadays known as Hermite-Padk approximation. Suppose we are given r Markov functions (or Stieltjes functions)


then our aim is to approximate these r functions simultaneously by rational functions. such that the approximation is of prescribed order near m. Type I Hermite- Pad6 approximation is to find a vector of polynomials A< 2 , . . . , A,- .) and a polynomial B,- such that the degree of A s , is less than or equal to n, - 1 and

Type I1 Hermite-Pad6 approximation is to find a polynomial PC of degree 161 and polynomials Q,-,, , . . . , Q,-,,. such that

Hence type I1 Hermite-Pad6 approximation is rational approximation of the functions f l , f2, . . . , f , with a common denomznator. For r = 1 type I and type I1 Hermite-Pad6 approximation reduces to ordinary Pad6 approximation of f l . It is well known that the denoniinator polynomials in Pad6 approximation to a Markov function (or Stieltjes function) are orthogonal polynomials for the measure p l . For multiple orthogonal polynomials there is a similar relationship: the polynomials A,- 1. A,- 2 , . . . , A,- , in type I Hermite-Pad6 approximation are precisely the type I multiple orthogonal polynomials for the measures ( P I , . . . , p,) and the common denominator P,- in type I1 Hermite-Pad6 approximation is the type I1 multiple orthogonal polynomial for the measures ( , u l , . . . . p,). Furthermore, the remaining polynomials BZ in type I approximation and Q,-, (3 = 1 , 2 , . . . , r ) in type I1 approximation can be obtained from the multiple orthogonal polynomials through the formulas

and

Combining (3.1) with (3 .5) gives a nice formula for the remainder in type I Hermite- Pad6 approximation in terms of the multiple orthogonal polynomials

and combining (3.2)-(3.4) with (3.6) gives a formula for the remainder in type I1 Hermite-Pad6 approxirnation


Hence for Hermite-Pad4 approximation one only needs to obtain the multiple orthogonal polynomials, from which all other quantities can be deduced.

3.2. Useful sys tems of functions. In order to get a system for which most multi-indices are normal, one needs to pose additional assumptions on the hlarkov functions ( f l , fi. . . . , f , ) or on the measures (p l . p2, . . . , y,). Two kinds of systems turn out to be quite useful.

DEFINITION 3.1 (see [An]) . An Angelesco s y s t e m consists of r measures on r pairwise disjoint intervals:

Angelesco systems are always complete systems and are the prototype for studying multiple orthogorlal polynomials on r pairwise disjoint intervals. It turns out that the zeros of type I1 multiple orthogonal polynomials have a very simple 10- cation property for Angelesco systems: there will be n, zeros of P,- on the interval A, for z = 1 ,2 . . . . , r . The polyrlornial PIT call therefore be factored as P,-(x) = P,, , (x) PI,, (x) . . . P,,, (x) and each P,,, is an ordinary orthogonal polynomial on A, for the varying measure n?+ PI,, (x) dp,(x).

For multiple orthogonal polynomials on one interval there is another useful system of functions.

DEFINITION 3.2 (see [N l ] ) . An AT s y s t e m consists of r measures on the same interval A such that each measure p, is absolutely continuous with respect to p = p1 and such that the Radon-Nikodym derivatives u, = dpJ/dp (with u1 = 1) are such that (1,s.. . . , x n 1 ' . u 2 , 2 ~ 2 . . . . , x7'2-' u2.. . . , u, , xu, , . . . , x ' ~ , - ~ u , ) is a Chebyshev system 011 A for every multi-index (n l , n2, . . . , n,).

Recall that a set of n linearly independent functions {41 ,4~ , . . . ,4,,) is a Chebyshev system on A if every linear combination C;=, akQr, has at most n - 1 zeros on A. For an AT system it thus follows that P,,,-1 (x) + PI,,-1 ( x ) u ~ ( z ) + . . . + P,,, (z)u, (x) has at most (61 - 1 zeros on A whenever Pr, is a polynomial of degree at most k . A specsal example of an AT systern is a Nzkzshzn s y s t e m [Nz] which is defined recursively as follows. Any hlarkov function

is a Nikishin system of order one on Al . A system f2 2 ) is a Nikishin system of order two on A2 if

where f 1 1 is a Nikishin system of order one on A l and Al n A2 = 0. In general the system (f, 1 , f, 2 , . . . , f l ,) is a Nikishin systen~ of order r on A, if

where f, -1 = 1 and (f, -1 1 , f,-1 2.. . . , f, , I ) is a Nikishin system of order r - 1 on A,- 1 with A, n A, -1 = 0. Nikishin systems turn out to be the prototype of systems of hlarkov functions on one interval and allow a rather detailed analysis. See. e.g.. Driver and Stahl [DSl ] [DSZ] for normality of Nikishin systems and


Bustamante and L6pez [BL] and Aptekarev [A31 for asymptotic results. Driver and Stahl [DSl] have shown that the type I1 multiple orthogonal polynomial Pi; for a Nikishin system of order r on A, has (61 simple zeros on this interval A, whenever n, 2 max(n,+l,. . . , n,) - 1 for every j = 1,2,. . . , r - 1. In particular this is true for the diagonal indices s'(n) given in (2.5).

Recently Gonchar, Rakhlnanov and Sorokin [ G R S ] have considered a mixture of Angelesco and Nikishin systems and have analyzed the asymptotic behavior of the corresponding Hermite-Padit approximants. The asymptotic behavior of the Hermite-Padit approximants for Angelesco and Nikishin systems was obtained in [GR] and [S5] . One of the important results is the asymptotic distribution of the zeros of the polynomials. For ordinary orthogonal polynomials the distribution of zeros is usually obtained by investigating an extremal problem for logarithmic potentials. For multiple orthogonal polynomials one needs to study an extremal problem in logarithmic potential theory for vector potentials.

4. Examples of multiple orthogonal polynomials

The very classical orthogonal polynomials of Jacobi, Laguerre, and Hermite have several possible extensions as multiple orthogonal polynomials. We will mention some of them. See also Aptekarev, MarcellAn and Rocha [AMR] for some Angelesco type multiple orthogonal polynomials.

4.1. Jacobi-Angelesco polynomials. Kalyagin [Kl] [KR] studied multiple Jacobi polynomials on two intervals [a, 01 and [O, 11, with a < 0. The measures are dpl(x) = IxIq(x - a)"(l - x)'~ dx on [a, 01 and dp2(x) = xnf(x - a)"(l - x ) ~ dx 011

[0, 11. with a , p, -y > -1. These intervals have only one point in common and this point will not influence the nature of the multiple orthogonal polynomials so that we can still consider this as an Angelesco system. There is a raising operator for the monic diagonal type I1 multiple orthogonal polynomials, namely

j - 1 ( 0 - 1 . j - 1 - , - I ) = - ( ~ y + ~ + - y + 2 n ) x ~ ~ ~ ( x - a ) " ~ ' ( l - x ) Pl,+,n+l (I),

and by using this fornlula repeatedly we obtain the Rodrigues formula

These polynomials satisfy a third order differential equation

where y(z) = pkI3 "(2). Observe that the coefficients of y"' and y" are polynomials independent of n but that the coefficient of y' depends on n, so that Sturm- Liouville theory of linear differential equations can not be used here. If we write

PJir = PAC:, ' ') and P2,1+l = pi';:,?, then we have a third order recurrence relation


for which

a + l 2 a + l 2 lim bZn = - + p, lim b2n+l = -

9 + -x2,

n-cc 71-30 9 3 4

lim c,, = -(a2 - a + l), n-x 81

4 4 lim dz , = - - x l ( x1 - a ) ( x ~ - l ) , lim d2n+l = - -x2 ( x 2 - a ) (x2 - 1 ) 71-x 2 7 n-x 2 7

where x l and 2 2 are the roots of the quadratic equation h l ( x ) = 0 , with h ( x ) =

x ( x - a ) ( x - 1 ) .

4.2. Jacobi-Pineiro polynomials. Pineiro [PI considered multiple Jacobi polynomials on [0, 11 with Jacobi weights

(4.4) dpl ( x ) = x n ~ ( 1 - x)O" d x , j = 1 ,2 , . . . , r ,

where a k > -1 for k = 0 , 1 , . . . , r and a, - a , $ Z whenever i + j and i , j # 0. Originally Pineiro only considered the case a0 = 0 but this more general case can easily be obtained without too much extra effort (see, eg . , [NS, p. 1621). If we set 6 = ( a 1 , . . . , a,) then we have r raising operators for the monic type I1 multiple orthogonal polynomials:

where Zj is the j t h standard unit vector ( 0 , . . . , 0 , 1 , 0 , . . . , 0 ) . Combining all these raising operators gives the Rodrigues formula

where the order in the product of r differential operators can be changed. For

r = 2 we put PZn = ~ i f f 2 . ~ ~ . ~ ~ ) ( a o 1 W . ~ 2 )

and P2n+l = P7L+l.n and we have the third order recurrence relation

for which the recurrence coefficients satisfy

lim b, = 4

7L-x !if7'

:I

i m dn = ($) n-x


4.3. Multiple Laguerre polynomials. In a similar way, Sorokin [S2] [S6] considered multiple Laguerre polynomials on [O, oo) with weights

(4.7) d p j ( x ) = x U ~ e - x d x , j = 1 , 2 , . . . , r ,

with a j > -1 for j = 1 ,2 , . . . , r and a , - a, @ Z for i # j . These polynomials are a limiting case of the Jacobi-Pineiro polynomials:

in a similar way as the Laguerre polynomials are a limiting case of the Jacobi polynomials. There are again r raising operators

and combining these raising operators gives the Rodrigues formula

For r = 2 we put P2, = L ~ A ' ~ ~ ) and P2n+l = ~?:;f*,?) to find the third order recurrence relation

xPn(x) = Pn+l(x) + bnPn(x) + cnPn-l(x) + 4 2 , - 2 ( x ) ,

with

bzn = 3n + a , + 1

bzn+l = 3n + a2 + 2

c2n = n(3n + a1 + a2)

czn+~ = (72 + 1)(3n + + a2) - an + 1

d2n = n ( n + a l ) ( n + a l - a 2 )

= n ( n + a z ) ( n + a 2 - a 1 ) .

Observe that

Another way to obtain multiple Laguerre polynomials on [0, oo) is to use the weights

(4.11) dp,(x) = xCYe-C~x dx , j = 1,2 , . . . , r ,

with a > -1 and c l ,cz , . . . ,c , distinct positive real numbers (see [ N S , p. 1601 where a = 0) . The raising operators are

and the corresponding Rodrigues formula is

and again the product of the differential operators can be taken in any order.


4.4. Jacobi-Laguerre polynomials. Sorokin [Sl] considered an Angelesco case containing a Jacobi and a Laguerre part. Consider the measures dpl (x) = (1 + x)" 1~1~'e-" dx on the interval [-I, O] and dp2(x) = (1 + ~ ) ~ x f l e - ~ dx on [0, m ) , with a, ,B > - 1, then the raising operator for the diagonal type I1 multiple orthogonal polynomial is

and applying this rule repeatedly gives the Rodrigues formula

4.5. Multiple Macdonald polynomials. For r = 1 the very classical orthogonal polynomials are those of Jacobi, Laguerre and Hermite. For multiple orthogonal polynomials there seem to be more classical families, which is at least suggested by the following example [VAY]. Consider the weight function

where a > -1 and p,(x) = x u / 2 ~ u ( 2 & ) (v 2 O ) , with K u the Macdonald function (modified Bessel function of the second kind). This AT system has a lot of properties that very classical polynomials usually have. The differential operator D is a lowering operator for the type I1 multiple orthogonal polynomials near the diagonal

and D, = x-"Dx" is a raising operator for the type I multiple orthogonal polynomials: putting QE ,,,, = A ~ , L , l L ) , l ~ u + AT T L , , L . n ) , 2 ~ u + ~ we have

Using the raising operators repeatedly gives a Rodrigues formula for the type I multiple orthogonal polynomials

Putting PL', = P& and P2rL+l = P:+l gives the recurrence relation

with

b, = ( n + a + 1 ) ( 3 n + a + 2 v ) - ( a + l ) ( v - I ) ,

c,, = n(n+a) (n+cr+v) (3n+2a+v) ,

d,, = n ( n - l ) ( n + a - l ) ( n + a ) ( n + c r + v - l ) ( n + a + v )


5. Irrat ional i ty

Multiple orthogonal polynomials turn out to be quite useful in proving irrationality of certain real numbers. The key to such irrationality proofs is the following lemma:

LEMMA 5.1. Let x be a real number. Suppose there exists integers p, and q, (n E N) such that

1. q,x - pn # 0 for all n E N, 2. lim,,,(q,x -p,) = 0.

Then x is irrational.

PROOF. Suppose x is rational so that x = plq with integers p and q. Then

By assumption (1) we know that q,p - pnq is an integer different from zero, hence Iq,p - p,ql 1. But then

and as n -+ m we find that this is in contradiction with assumption (2). Hence x cannot be rational. 0

This lemma shows that if one can find good rational approximants p,/q, such that jx - p,/q, 1 = o(l/qn), then x has to be irrational. Such irrational numbers can therefore be approximated very well by rational numbers. Constructing rational approximants to x can be done using Pad6 approximation or Hermite-Pad6 approximation, especially when x is the value of a Markov function f (2) with z far enough away from the support of the measure. This is the case for some values of the zeta function

in particular we have for all positive integers

dx. j !

We can use that information and multiple orthogonal polynomials for the measures dpj (x) = (-l)J logj(x) dx on [O,1] for j = 0 ,1 ,2 to prove the irrationality of C(3). When Apkry [Ap] first announced the proof of irrationality of ((3), the proof was considered as rather mysterious. Later Beukers [Bl] [B2] showed how Ap6ry's approximants can be obtained using Hermite-Pad6 approximation.

THEOREM 5.2 (Ap6ry). C(3) is irrational.

PROOF. In order to obtain good rational approximants to <(3) we will use Hermite-Pad6 approximation to the three Markov functions

logx 2 dx, f3(z) = 1' dx, Z - x Z - x


and use that f 3 ( 1 ) = ( ( 3 ) . We want to find a vector of polynomials (A,, , B,), where A, and B , are polynomials of degree n , and polynomials C, and D, such that

Observe that (5.2) is a type I approximation problem for the system ( f l , f 2 ) and (5.3) is a type I approximation problem for ( f 2 , f 3 ) , whereas the combination (5.2)- (5.3) is a vector type I1 approximation problem with common vector denominator (A, , B,,). The solution of this problem in Hermite-Pad6 approximation is given by the generalized polynomial A, ( x ) - Bn (x) log x for which the following orthogonality relations hold:

1

(5.4) ( A ( x ) - B , , ( x ) l o g x ) x k d x = 0, k = 0 , 1 , . . . , a - 1

(5.5) l(A,(x)-~,(x)logx)log(x)xkdr = 0, k = 0 , 1 , . . . , n - 1 , 0

together with the extra condition that A,(1) = 0. Let P, be the Legendre polynomials of degree n on [0,1] ,

then the Mellin convolution

is of the form A,,(x) - B,(x) logx with A,(1) = 0. Taking the Mellin transform gives

and for s = 0,1 , . . . , n - 1 this gives the orthogonality conditions in (5.4). Taking derivatives gives

1 1 /' Frl(x)x.' logx dx = 2 1 Pn(x)xs dx 1 P,(x)xs log x dx , J o

and for s = 0 , 1 , . . . mainder in (5.3) is

and hence, choosing

, n - 1 this gives the orthogonality conditions (5.5). The re-

1 - 1 An ( x ) - Bn ( x ) log x log x dx ,

2 - 2

z = 1 in (5.3) gives

L I A , (3) - &(x) logx m n ( 1 ) c ( 3 ) - ~ ~ ( 1 ) = - log x dx.

1 - x

Using the representation (5.7), the Rodrigues formula for P, and


gives

so that 2Bn(1)((3) - D,(l) > 0. A simple estimation of xyz( l - x) (1 - Y) (1 - z ) / ( l - (1 - xy)z) on the unit cube gives the upper bound

Explicit calculation gives

and

1 ""(1) - *"(XI log(x) dx + - Bn(x) log2 dx 1 = - 1

l - x I1 1 - x

~ f ) - Hj2)

j - k ' k=O 1 = 0

j # k

where ~ f ) = l/j2. Clearly B,(1) is a positive integer, but D,, (1) contains rational terms. However, if we multiply D n ( l ) by a common multiple of the nurnbers 2 9 3 " , , . . . , n%hen we would get an integer. Taking d, = lcm(1,2,3,. . . , n) (least common multiple) then

Taking the nth root and using the asymptotic property

lim d;In = e, n-cc

then gives, with p, = 2diBn(1) E N and q,, = diD,(1) E Z

hence Lemma 5.1 implies that ((3) is irrational.

6. Transcendence

Historically, Hermite introduced the notion of Hermite-Pad6 approximation to prove the transcendence of e. The basis lemma for proving transcendence is a generalization of Lemma 5.1, which goes as follows:

LEMMA 6.1. Let x be a real number. Suppose that for every posztzve znteger m E N and for all zntegers ao, a l , . . . , a, E Z we can find zntegers po ,, , p l . , . . . ,p,., (n E N) such that

1. C;ko akpk ,, # 0 for all n E W.


2. limn,,(po.nxk - ~ k . ~ ) = 0 for k = 1 ,2 , . . . , m .

Then x is transcendent.

PROOF. Suppose that x is algebraic, then there exists a positive integer m and integers ao, a l , . . . , a m such that CF=o akxk = 0. By adding and subtracting we find

We know by assumption (1) that the integer CTxO akpk., is different from zero, hence in absolute value it is at least 1. As a consequence

I m I

Taking n -+ co, then assumption (2) gives a contradiction. Hence x has to be transcendental. 0

Proving transcendence can therefore be done constructively by producing good simultaneous approximants to consecutive powers of x. Type I1 Hermite-Pad6 approximants can then be used in case x3 = f j (z) for a complete system of Markov functions ( f l , fi, . . . , f,). Hermite first showed that (ecl", ec2", . . . , eCrZ) is a complete system for Hermite-Pad6 approximation near zero (rather than near co, which we described in this paper) whenever all ci are distinct complex numbers. He then used the type I1 Hermite-Pad6 approximants for the system (eZ, e2", . . . ,em") to produce simultaneous rational approximants satisfying the conditions of Lemma 6.1. See [NS, pp. 130-1311 or [VA2] for details of this proof.

References

[An] M. A. Angelesco, S u r d e u x ex t ens ions des fract ions con t inues alge'brzques, C.R. Acad. Sci. Paris 18 (1919), 262-263.

[Ap] R. Apkry, Irrat ional i t6 d e C(2) e t C(3), Ast6risque 61 (1979), 11-13.

[All A. I. Aptekarev, A s y m p t o t i c s of s imul taneously orthogonal polynomials in t h e Angelesco case, Mat. Sb. 136 (1988), 56-84 (in Russian); Math. USSR Sb. 64 (1989), 57-84.

[A21 A. I. Aptekarev, Mult iple orthogonal polynomials, J . Comput. Appl. Math. 99 (1998), 423-447.

[A31 A. I. Aptekarev, S t rong asympto t i c s of mul t ip le orthogonal polynomials for Nzkishin sys- t e m s , Mat. Sb. ( to appear).

[AMR] A. I. Aptekarev, F. Marcellin, I. A. Rocha, Semiclass ical multzple orthogonal polynomials and the properties of Jacobi-Bessel polynomials , J . Approx. Theory 90 (1997), 117-146.

[AS] A. I. Aptekarev, H. Stahl, A s y m p t o t i c s of Hermi t e -Pad6 polynomials , in 'Progress in Approximation Theory' (A. A. Gonchar, E.B. Saff, eds.), Springer-Verlag, Berlin, 1992, pp. 127-167.

[Bl] F. Beukers, A n o t e o n the irrationalzty of ((2) and C(3), Bull. London Math. Soc. 11 (1979), 268-272.

[BZ] I?. Beukers, Pad6 approx iman t ions in n u m b e r theory , in 'Pad6 Approximation and its Applications', Lecture Notes in Mathematics 888, Springer-Verlag, Berlin, 1981, pp. 90- 99.

[BE] P. Borwein, T. Erdblyi, Polynomia l s and Po lynomia l Inequalities, Springer-Verlag, Berlin, 1995.

[dB] M. G . de Bruin, S imul taneous Pad6 approx iman t s and orthogonality, in 'Polyn8mes Or- thogonaux et Applications' (C. Brezinski et al., eds.), Lecture Notes in Mathematics 1171, Springer-Verlag, 1985, pp. 74-83.


[BL] J. Bustamante, G. Lbpez, Hermzte-Pad6 approximatzon for Nikishzn systems of analytic functions, Mat. Sb. 183 (1992), 117-138 (in Russian); Math. USSR Sb. 77 (1994), 367-384.

[DSl] K. Driver, H. Stahl, Normalzty i n Nzkzshin systems, Indag. Math. (New Series) 5 (2) (1994), 161-187.

[DSZ] K. Driver, H. Stahl, Szm,ultaneous rational approxzmants to Nzkishzn systems, I, II, Acta Sci. Math. (Szeged) 60 (1995), 245-263; 61 (1995), 261-284.

[GR] A. A. Gonchar, E. A. Rakhmanov, On the convergence ofszmultaneous Pad6 approximants for systems of Markov type functions, Trudy Mat. Inst. Steklov 157 (1981), 31-48 (in Russian); Proc. Steklov Math. Inst. 3 (1983), 31 50.

[GRS] A. A. Gonchar, E. A. Rakhmanov, V. N. Sorokin, Hermzte-Pad6 approxzmants for systems o f Markov-type functzons, Mat. Sb. 188 (1997), 33-58 (in Russian); Russian Acad. Sci. Sb. Math. 188 (1997), 671-696. V. A. Kalyagin, O n a class of polynomzals defined by two orthogonality rela,tions, Mat. Sb. 110 (1979), 609-627 (in Russian); Math. USSR Sb. 38 (1981), 563-580. V. A. Kalyagin, Hermite-Pad6 approximants and spectral analyszs of nonsymmetric operators, Mat. S b . 185 (1994), 7 9 1 0 0 (in Russian); Math. USSR Sb. 82 (1995), 199-216. V. A. Kaliaguine, A. Ronveaux, O n a system of classzcal polynomials of simultaneous orthogonalzty, J. Comput. Appl. Math. 67 (1996), 207-217. R. Koekoek, R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Delft University of Technology, Report 98-17, 1998. Available on-line a t http://aw.twi.tudelft.nl/-koekoek/research.html E. M. Nikishin, A system of Markov functzons, Vestnik Mosk. Univ., Ser. I (1979), no. 4, 60-63 (in Russian); Moscow Univ. Math. Bull. 34 (1979), 63 66. E. M. Nikishin, O n simultaneous Pad6 approximants, Math. Sb. 113 (1980), 499-519 (in Russian); Math. USSR Sb. 41 (1982), 409-425. E. M. Nikishin, V. N. Sorokin, Rational Approximations and Orthogonality, Translations of Mathematical Monographs 92, Amer. Math. Soc., Providence, RI, 1991. L. R. Pineiro, O n simultaneous approximations for a collection of Markov functions, Vest- nik Mosk. Univ., Ser. I (1987), no. 2, 67-70 (in Russian); Moscow Univ. Math. Bull. 42 (2) (1987), 52 55. V. N. Sorokin, Szmultaneous Pad6 approxzmants for finzte and infinite zntervals, Izv. Vyssh. Uchebn. Zaved., Mat. (1984) no. 8 (267), 45-52 (in Russian); J . Soviet Math. 28 (1984) no 8, 56-64. V. N. Sorokin, A generalitatzon of classzcal orthogonal polynomzals and the convergence of simultaneous Pad6 approxzmants, Trudy Sem. Im. I. G. Petrovskogo 11 (1986), 125-165 (in Russian); Soviet Math. 45 (1989), 1461-1499. V. N. Sorokin, A generalization of Laguerre polynomials and convergence of szmultaneous Pad6 approximants, Uspekhi Mat. Nauk 41 (1986), 207-208 (in Russian); Russian Math. Surveys 41 (1986), 245 246. V. N. Sorokin, Convergeme of szmultaneous Pad6 approxzmations to functzons of Stzeltjes type; Izv. Vyssh. Uchebn. Zaved., Mat. (1987), No.7 (302). 48 56 (in Russian); Soviet Math. 31 (1987) no. 7 , 63-73. V. N. Sorokin, Convergence of simultaneous Pad6 approxzmants for a class of functzons, Mat. Sb. 132 (1987), 391400 (in Russian); Math. USSR Sb. 60 (1988), 385 394. V. N. Sorokin, Szmultaneous Pad4 approximatzon for functzons of Stieltjes type, Sib. Mat. Zh. 31, no. 5 (1990), 128-137 (in Russian); Sib. hlath. J. 31, no. 5 (1990), 809 817. G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23, fourth edition 1975.

[VAl] W. Van Assche, Orthogonal polynomials i n the complex plane and on the real line, Fields Institute Comm. 14 (1997), 211-245.

[VA2] W. Van Assche, Approximation theory and analytic number theory, in 'Special Functions and Differential Equations' (K. Srinivasa Rao et al., eds.), Allied Publishers, New Delhi, 1998, pp. 336-355.

[VAY] W. Van Assche, S. B. Yakubovich, Multiple orthogonal polynomials associated with Mac- donald functions, (manuscript).


DEPARTMENT OF MATHEMATICS, KATHOLIEKE UNIVERSITEIT LEUVEN, CELESTIJNENLAAN 200 BUILDING B, B-3001 HEVERLEE, BELGIUM

E-mail address: walterQwis . kuleuven. ac. be


Reduction of Continued Fractions of Formal Power Series

Alfred J. van der Poorten

ABSTRACT. A formal power series F (X) in Q((X-I)) normally has almost all its partial quotients polynomials of degree 1, and with rational coefficients growing in complexity at a furious rate. We mention some classes of atypical examples. In particular we recall the case of power series representing an algebraic function, where 'atypicality' corresponds to a classical problem.

In the generic case it seems one should consider series over Fp rather than over Q . Obviously, if the partial quotients of the series over Q all have good reduction at p then the continued fraction over Fp is simply the reduction of the original expansion. But if some partial quotients have bad reduction at p it is less clear how to effect the reduction. However, we show that there is a clear sense in which the convergents in characteristic p are precisely given by the reductions mod p of the convergents in characteristic zero.

1. Introduction

We remark on general properties of simple continued fraction expansions of formal power series (hereafter omitting the qualifier 'simple'). We explain various basic properties of the expansions. For example, a little experimentation with 'random' formal power series, say with coefficients 0 , 1, and -1 only, suggests that - unless there are large gaps of zero coefficients - all, certainly almost all, their partial quotients are of degree one, and that the coefficients of those partial quotients increase in complexity at an exponential rate. It follows that,, generically, the convergents to formal power series are rational functions with coefficients growing at a rate compounding that of the partial quotients. A consequence is that those convergents are not helpful in the study of diophantine properties of values taken by a series in its domain of convergence. Of course there are atypical cases; we mention some classes of examples. Indeed, we briefly suggest which cases are 'unexpectedly' atypical and thus interesting.

An evident consequence of the explosive growth of the numerical coefficients is that computer programs computing such expansions soon choke - they slow down severely. One might choose to avoid that by studying the continued fraction expansions of the series reduced modulo p , say for some collection of primes p , thus 'confining' and taming the coefficients of the partial quotients. We formulate and prove a 'Reduction Principle' relating the reduction mod p of a continued fraction

1991 Mathematzcs Subject Classzfication. Primary 11570, l lA65, 11568. Supported in part by a grant from the Australian Research Council.

@ 1999 American hlathernatical Society

343

344 ALFRED J. VAN DER POORTEN

expansion to the original expansion in characteristic zero. In principle, having done this for sufficiently many primes, one can then retrieve the characteristic zero expansion by the Chinese Remainder Theorem. This is not entirely straightforward; generically the continued fra,ction expansions as such have bad reduction at all primes. One cannot just reduce the expansion to obtain the expansion of the reduction. Seemingly, one must reduce the series mod p and compute the expansion of the reduction afresh. However, our reduction principle shows that one may so view the convergents that they have meaningful reduction mod p and that then the reduction of the convergents yield all convergents of the reduced power series.

1.1. Basics. The familiar continued fraction algorithm, normally applied to real numbers, is readily applied analogously to formal Laurent series CT=-, g h ~ - h

in a variable X p l . One takes the 'integer part' of such a series to be its 'polynomial part' gp,Xm + . . . + g _ l X +go. Then the partial quotients are polynomials in X , and we learn that continued fraction expansions

with partial quotients polynomials of degree at least 1 in X (perhaps excepting ao) and defined over some field, apparently converge to formal Laurent series in X-I over that field.

Specifically, given a Laurent series F ( X ) - we will assume it not to be a rational function unless the contrary is clearly indicated - define its sequence (Fh)h20 of complete quotients by setting FO = F, and Fh+l = l / (Fh - ah(X)) . Here, the sequence of partial quotients of F is given by ah = [Fh] where 1 1 denotes the polynom?al part of its argument. Plainly we have

Only the partial quotients matter, so such a continued fraction expansion may be conveniently detailed by [ao . a, , a2 , a3 , . . . ] = [a() , a1 , a2 , . . . , a,, , F h + l ] .

The truncations [a" , a l , . . . , ah ] are rational functions ph/qh. Here. the pairs of relatively prime polynomials ph(X), qh(X) are given by the matrix identities

and the remark that the empty matrix product is the identity matrix. This alleged correspondence, whereby these matrix products provide the convergents ph/qh, may be confirmed by induction on the number of matrices on noticing the definition

[ao , a1 , . . . , ah] = a" + l / [ a l , . . . , ah] , [a"] = a u .

It follows that the continuants qh satisfy degqh+l = degah+l + degqh. Incidentally, we also see, by transposing the matrix correspondence, that

[ah , ah-I , . . . , a1 ] = qh/qh-,, for h = 1,2 , . . . .

REDUCTION OF CONTINUED FRACTIONS OF FORMAL POWER SERIES 345

The matrix correspondence entails ph/qh = Ph- I /qh-I+(- 1) h-l/qh-lqh whence, by induction, F = a0 + ~ ~ = l ( - l ) h - l / g h - l q h , and so

displaying the excellent quality of approximation to F provided by its convergents.

PROPOSITION 1.2. Let p , q be relatively prime polynomials. Then

zf, and only if, the rational function plq is a convergent to F

PROOF. The 'if' part of the claim has already been remarked, so we may take Iz so that deg qh-1 5 deg q < deg qh , and note that supposing plq is not a convergent implies that q is not a constant multiple of qh-1. Then phqh-I - qhph-1 = *1 entails there are nonzero polynomials a and b so that

and SO qF-p = a(qh - IF-ph - l )+b(qhF-ph ) . Now suppose that the two terms on the right are of different degree, deg a - deg qh and deg b - deg qh+l, respectively. In that case plainly deg(qF - p) > deg(qh-lF - ph-1) > deg(qhF - ph) , confirming that the convergents provide the locally best approximations to F .

To verify the claim that the degrees of the two terms are different, notice that deg aqh-I = deg bqh , otherwise deg q < deg qh is not possible, so

dega - degqh = degb - degqh-1 > degb - degqhtl.

Moreover, dega - degqh = deg(qF - p ) . So it remains to confirm that

deg a - deg qh 2 - deg q.

But that is plain, because dega must be at least as large as degqh - degqh_l.

1.2. Normality. It's actually quite easy to see why, for formal power series over an infinite field, all partial quotients are almost always of degree one. We need only notice that a remainder C h > a h x P h has a reciprocal with polynomial part of degree greater than one, and thus gives rise to a partial quotient of degree greater than one, if and only if a1 = 0. Moreover, the partial quotient is a r l x - a2a,' if a1 # 0 , and the next remainder is (a; -ala3)a,"~-' +terms of lower degree in X .

A little more precisely, one observes that the sequential remainders have leading coefficient some determinant, and hence some multivariate polynomial, in the coefficients of the formal power series. But such a polynomial is 'nonzero "almost always", or "with probability one" ' - [9], p.375. The matter is considered in ex- tenso by Knuth [9] in the context of his discussion of the Euclidean algorithm for polynomials over a field. Moreover, that discussion also shows that the 'explosive growth' of the coefficients of the partial quotients is precisely the better known extraordinary growth of the coefficients in performing the Euclidean algorithm on a pair of polynomials over the integers Z.

Over the finite field F, , on the other hand, the nonvanishing of the coefficient of X-I of all remainders is as unlikely as is the nonappearance of the digit 0 in the base p expansion of a random real number. Thus we should expect that the reduction mod p of a formal power series almost always has partial quotients of


degree greater than one and that almost always the original sequence of partial quotients nowhere has good reduction.

Power series with substantial gaps are of course not generic. To assist we mention

PROPOSITION 1.3 (Folding Lemma [ll]). W e have

PROOF. Here - denotes the 'correspondence' between 2 by 2 matrices and continued fractions. We have

as alleged. Moreover, [x - qh-I /qh ] = [x , -& ] by Proposition 1.1. 0

The adjective 'folding' is appropriate because iteration of the perturbed symmetry w --+ w,x, -%i yields a pattern of signs on the word w corresponding to the pattern of creases in a sheet of paper repeatedly folded in half; see [6].

The folding lemma makes it easy, given the expansion of a partial sum, to adjust a continued fraction expansion for an appended term if the intervening 'gap' is wide enough. Specifically if the partial sum has degree -n one can readily append a term of degree -2n or less. Of course if the degree of the appended term is much less, that introduces a partial quotient of correspondingly high degree.

As an application of the lemma: the continued fraction expansion of the sum

i~givensequent~ially by l + X - ' = [ l , X ] , l + X - ' + X - " [ l , X , X , X ] , 1 + X- ' + X-3 + X P 7 = [ l , X , X , X , fT , X , X , fT], . . . , where the addition of each term is done by a 'fold' with x = - X ; see [15].

In any case, other than when the power series has blatantly large gaps, our two opening remarks combine to explain the claim that one may expect a formal power series, say with integer coefficients, to have partial quotients of degree one, that the continued fraction expansion will have bad reduction at all primes, and - noting the shape of the coefficient of X p l of the 'next' remainder - that the coefficients of the partial quotients will compound in complexity at an exponential rate.

1.3. Some infinite products. Originally, I met the phenomena discussed here in the context of studying certain of the infinite products

For k > 3 the truncations Gk.+ ( X ; n) of the infinite product Gk,* (X) are readily seen to be convergents of Gk,*. It also turns out to be easy to see that in the cases k , +, with k even not 2 , they are every second convergent of G k . + . It follows that every ~ a r t i a l quotient of the G2k.+ has integer coefficients.


In particular, there is - a degenerate case - Euler's product

G2,+(X) = n ( 1 + x - ~ ~ ) = l/(l - X-') = [ I , X - I ] . h>O

The remaining cases are very different. It turns out that for the G2k.-, and for all the GZk+l.+, the continued fraction expansion exhibits the generic behaviour mentioned in our introduction above. For k = 3 , the partial quotients that arise from the fact that the truncations of the products all are convergents also happen to have degree one. For k > 3 they do not, and then neither do the partial quotients propagated by the functional equations

All other partial quotients are of degree one, except in the case 2, - . In summary, the exceptional cases among the Gk.+ are the Gzk,+ , in having all

their partial quotients specialisable, that is, with integer coefficients and thus with good reduction at every prime - compare the G2k,- for k 2 2; and the example G 2 , , in having a partial quotient of degree more than one. All other cases that have partial quotients of degree other than one can be credited to 'blatant' gaps in the power series.

2. Reduction of Formal Power Series

If p never occurs in a denominator of a partial quotient then the expansion has good reduction, and we can just reproduce it reduced modulo p. But it's not clear what should happen when there are partial quotients with bad reduction at p .

2.1. An example of reduction mod p. Fortunately, it's easy to begin to answer that question by computing an example.

G2,- (X) = n (1 - x-2") =

h 2 0

[ I , - X + 1 , - ; x - 1 , 2 ~ " 2 ~ + 4 , -;x, 2 x 2 + 2 x , ;x-1, x + + , ;x+y ,

. . . . . omitting half a dozen partial quotients to improve the pagebreak . . . . . . ,


Contrary to general forecast, there is a partial quotient of degree other than one; it is generated, in effect, by a numerical accident. All other partial quotients are indeed of degree one, other than the images of the said partial quotient of degree 2 under the functional equation. Of course the detail reported above has no particular meaning other than as vivid illustration of the explosive growth of the coefficients of the partial quotients.

For example, over IF3 we find that

As suggested, to the extent that the original expansion has good reduction the new expansion is its reduction; whilst the first term with bad reduction 'collapses' to a term of higher degree. Beyond that term, the expansion is not immediately recognisable in terms of the original. Notice that for this example the functional equation entails that once the 'collapse' yields a partial quotient of high degree then partial quotients of increasingly high degree are propagated throughout the expansion.

2.2. Reduction Principle. We will want to use p to denote a prime; so our convergents will here be x ly . As above, given a series F , we denote its sequence of partial quotients by ( a h ) , and of its complete quotients by (Fh) .

The idea is to return to a first principles genesis of the sequence of polynomials (yh) yielding the convergents to F . We know that the convergents xh/yh are characterised by the property

So the series F characterises a strictly increasing sequence (dh) of integers dh = deg yh. Then the yh are those polynomials, of degree dh respectively, so that the Laurent series y h F has no terms of degree -1, -2, . . . , nor -dh . There is now nothing to stop us from normalising the polynomials yh so that each has integer coefficients not sharing a common factor. Presuming that F has good reduction at p , that entails a normalisation for the xh that also gives the xh good reduction at p . Note that the yh are certain constant multiples of the continuants qh provided by the matrix formulaire. Now consider this story in characteristic p . In characteristic - p we can use the same words; we mark all reduced quantities with a .

THEOREM 2.1 (Reduction Principle). The distinct reductions of the yh yield all the convergents of F .

PROOF. Certainly, each yh yields a convergent to F , because

deg(yhF - xh) < - deg yh implies that deg( y h F - xh ) < - degyh < - d e g K .

However, some of the yh may coincide. Denote representatives of the distinct yh by -- y h ( ~ ) , yh(l), . . . , Yh(l), . . . , where each h ( j ) is maximal; that is yh(j) = yh(j)-1=


The last inequality informs us that the corresponding next partial quotient of F, let's call it bj+l , has degree at least deg yh(,)+, - deg Yh(J). But

where we recall yh(,) = yh(, - 1)+1 , and that by the formalism yh(-l )+1 = 1 SO that yh(-l)+l is of degree zero.

However, it is plain - say again from the matrix formulaire, that

It follows that the inequalities above all are equalities, that is, deg yh(, - =

d e g y h ( ~ - i ) + ~ and degyh(j)+i - degyh(,) = degbj+l, and the yh(,) must account for all the convergents of F as claimed. 0

In [2] it is shown that all partial quotients of the product

G3,+ ( X ) = (1 + X-"I) = h=O

have degree 1 . Indeed, the partial quotients do all appear to be linear, and their coefficients

grow in complexity at a furious rate - the 30th partial quotient is - 1 3 7 4 3 8 9 5 3 4 7 2 0 ~ - 13743895347200 .

1573711 1 456376219 1

the 50th is

The argument of [2] uses the functional equation

G3,+(X; 72 + 1) = ( X + 1)/X . G3.+(x3; n)

relating the partial products. It 'grows' the continued fraction expansion first by replacing X by x3 and then multiplying the resulting expansion by ( X + l ) / X , using ideas suggested by Raney [17]. Even then, it remains necessary to show there is no 'accidental' degeneracy, and that is done by showing that, modulo 3 , the process is periodic. Underlying that is the observation that the reduction of G3,+(X) over IF3 is (1 + x - ' ) - ' / ~ , a quadratic irrational with periodic continued fraction expansion.

However, Cantor [5] remarks that the normality of the continued fraction expansion of G3.+ is already an immediate consequence of the fact that all the partial quotients of (1 + x p l ) - l / ' over IF3 are of degree 1. The Reduction Principle gen- eralises Cantor's argument.

350 ALFRED J . VAN DER POORTEN

Indeed, it is the simplest case of the Principle that if degyh = h for all h then the same is true for the original function, that is: deg yh = h for all h . But it is easily confirmed that degyh = h for all h for (1 + X-')-'/' over IF3; specifically

(1 + ~ - l ) - l ' ~ = [ l , X , -X + 1 , X - 1 1 . So of course dh = deg yh = h is true for &,,(l+ - X-3"), as Cantor pointed out.

3. Series with a Specialisable Continued Fraction Expansion

As remarked, examples with large gaps - doubling the exponent or more - are not interesting. In that spirit, the example of [16]:

ts interesting. Here the sequence of exponents (F,,) of the series is defined by the recurrence relation Fh+2 = Fh+l + F h and the initial values F2 = 1 , F3 = 2; and the point is that the golden ratio (1 + &)/2 is less than 2. Notwithstanding that, in [16] we succeed in proving the validity of the expansion by contorted applications of the Folding Lemma (Proposition 1.3).

Here the expansion is speczalisable, in the sense that we may replace X by a positive integer, other than 1, and readily obtain a valid numerical continued fraction expansion - because it is a simple matter to modify nonpositive partial quotients in a continued fraction expansion. Of course, this is the same thing as the expansion having integer coefficients - and thus good reduction at all primes.

One wonders which other recurrence relations produce sequences of exponents proving a specialisable continued fraction expansion. Experiment suggests that this class may well be restricted to certain sequences (Gn.h) satisfying the relation Gn,h+ntl + Gn.h = 2Gn,h+, . Those are translates by a constant of sequences satisfying

Gn.h+n = Gn.h+n-1 + Gn.h+n-2 + . . . + Gn,h . The case n = 2 of course includes the sequence of Fibonacci numbers.

4. Power Series with Periodic Continued Fraction Expansion

It is plain that a periodic continued fraction cannot be generic - periodicity entails that the partial quotients of a periodic expansion cannot ~ust~ain explosive growth of their coefficients. However, it is easy to confirm and well known that periodicity is possible only for quadratic irrational functions. Even then, as I remark below, periodicity of continued fraction expansions of formal power series is a quite unusual phenomenon.

One knows that a number y has a periodic continued fraction expansion if and only if it is a real quadratic irrational. Indeed, such a number generates a real quadratic number field K , and such a field contains a nontrivial unit, say p - qy, where p , q # 0 are integers. Say y has norm N and trace T. Given a unit p - qy , one considers the matrix


noting that the matrix A1 has such deconlpositions for integers n o . . . . , 0 ,

bccause it is unirnodular with integer entries (and noting that it riceds a rrionient's thought to see that that those entries indeed are integers). Whatever. I claim that

PROPOSITION 4.1. The nmmber. -, has the periodic contznu,ed fraction c:rpansbon 3 = [ G , a ( , . . . . a,.-, 1 .

PROOF. By the matrix correspondence. the allegation is

In other words. the claim is

7 = PY - N Y

cl^u+p-Tq But that is just Y(-,' - T y + N) = 0 and, recalling q # 0 . yields the allegation. 0

Note however that the integers a , may he tionpositive: mleed it is an iiistructive exercise to confirm - assuming p arid q are positive - that all the a , may bc chosen positive if arid only if > 1 and its co11,jugate 7; satisfies -1 < 7; < 0. Nonetheless, various simple transformations. specifically the rules

readily adjust the expansion to having positive entries. These transformations maintain periodicity but may of course introduce a preperiod and radically change the period length.

Conversely, given a rlusnber with periodic continued fraction exparlsion it is straightforward to confirm that its period yields a unit in thr. quadratic siunhtr field it generates.

Somehow it will seem iriore natural to use .I 111 place of X in the sequel. The story just told is of course much the same for the continurd fraction expansion of a quadratic irrational y = y(1) over a field K(.r) of rational functions. Suppose y(x) has norm lV(x) and trace T ( x ) . Exactly as above. the existence of a unit p(7) - q(x)y entails that there are polynomials .,(.I) given by a decomposition

so that y(x) = [ao(x) , a , (x) : . . . . a,.-, (T) 1 . Here. a polynomial partial ql~otient is inadmissible only if it is constant.

Now suppose that y" = D ( r ) is a polynomial, riot a perfect square in K[.r]. Just as in the numerical case, prcsuniing the exist,encc of a unit p(x) - q(.r)y shows t,hat has an expansion

with each of the a , of positive degree. hloreovcr, because the trace T(x) vanisl~es. it follows that the word ( 1 1 (3.). . . . , a , - (x) is a palindrorrie. Not ice, incidentally.

2 that the equation (JI(T))' - ( q ( ~ ) ) y = &1 entails that D has even degrec. and leading coefficient a square in K.


But these ~ondit~ions do not suffice for the existence of a unit p(x) - q(r)y: unless K is a finite field. In brief, in number fields one proves the existence of nontrivial units by applications of the box principle. For infinite fields K , though. such arguments fail for function fields over K because there are i~lfinit~ely many polynomials of each degree. In all. given a generator y of a quadratic extension it is unusual for t,here to he a unit of the shape p ( r ) - q(r )y . Indeed, with y" D(x) suppose there exists such a nontrivial unit. It being a unit. its divisor class on t,he Jacobian of the curve y2 = D(x) is supported at the two divisors at infinity: say x+ and o o . Thus for some positive integer n , n ( m + - m - ) is the divisor of a function. I11 other words. the Jacobian has a torsion point of order n .

Not much is as yet known about such t,orsion points on Jacobians of arbitrary genus g - corresponding t,o deg D ( x ) = 2g+2 ([8] is a useful current reference with helpful examples and extensive guide to the literature). However, if K = Q and g = 1 we have an elliptic curve over Q and then a theorem of hIazur [lo] details all possibilities for rational torsion. In particular, the highest torsion possible is 12.

IVe saw above that has a preperiod of length one: it follows that the case 71. = 12 should yield a period of length 11. For a more leisurely explanation see [3]. I am not aware of such a periodic continued fraction expansion ever having been ~nentiorled in the literature arid take this opport,unity to display a computation of Xuan Chuong Tran detailing up t,o isomorphism the entire class of Q-curves

with a torsion point of order 12 at o o (and where m+ is the 0 for addition on the curve). Specificall>-, one has such a curve for every rational t # 0, f , 1 if

Before detailing the expansion of y = \/xJ + cr" dx + e , we had best admit that. in our remarks above p(x) - q(x)y is defined as a 'unit' already if its norm is in KX : our criterion does not demand that its norm be h l . Below we display the first twelve partial quotients of the expansion of y , but just the first half of its ptlriod - which is of length 22; twice as long as suggested. What has occurred is that the 11-th convergent plo/qlo yields a 'unit' plo(x) - qI0(x)y with norm

+ c(t) = 2t(2t - 1)'(3t2 - 3t + l ) ' / ( t - 1)" in Q X . Suppose [ a o . w , a l l ] denotes the first 12 partial quotients of y . Then the complete expansion of y - given by the

2 2 i t unit (plo(.r)-qlo(.r)y) / (c( t )) i s o n t h e o n e h a n d [ a o , w . a l l , a l l , w , O . a o ] .

i by the known symmetry, on the other hand [uo , w , a l l , ao/c(t) , c ( t ) Z , 0 , a0 1 . h\- our present remark on yuasz-units. Here the word f;; is the string of partial q;oticGs in reverse order and [c ( t ) ] 1s the continued fraction o f c(t) [ ; one of course recalls that d[n , h , c , . . . ] = [ ad , h/d , cd , . . . ] . It's now plain that w must itself be c-sy~mnetric - in the sense that c z = f;;; : as one readily confirms. By the way. these phcnornena are present in the numcrical case. It's just that then multiplication by \om? constar~t c is rather more difficult to notice; cf [12].


a,, ( 2 )

Take t = 2 as an instance. Of course, this example is detailed purely for fun, to show it can be done. But the principle it exemplifies is important in the context of our discussiorl of formal power series with 'exceptional' continued fraction expansion: with good reduction almost everywhere. That phenomenon is rare. It occurs only when the Jacobian of the corresponding curve has a torsion point at infinity.

354 ALFRED 3. VAN DER POORTEN

The issue is classical and motivatmed hbcl's st,udy of what we now call abclia~i integrals. In the present,. hyperelliptic. case t,he quest,ion concerns so-c:allrd quasi- ellipt,ic int,egrals - for which there is some polynomial f ( s ) so that the differential f ( x ) d x / y is of the shape d 1 1 / 7 ~ for some algebraic fur~ction ZL - - in othcr words so t,hat an int,egral t,hat morally should yield some inverse of a nasty gerieralised inverse c,lliptic function in fact provides just t,he logaritliin of an algebraic funct,ion. In fact. I take this explanation from [I]. with deg D = 2y + 2 t,he curve ?j2 = D ( r ) has gems g and thc holomorphic differential d.x./y has divisor ( g - 1 ) (m+ + c- ) . Because the tliffrrential d u l u has just simple poles - at the zeros and poles of (1

one must have deg f = y and ZL can have zeros or poles orill\- at x+. In ot,ller words. 11 must be a nontrivial unit,.

hloreovcr. since p2 - q 2 D = c , on differentiating with respect t,o s we have

so q l p 1 p . whence q p 1 . Then the polynornial f = p l / q is of degree y and one readily verifies that with 71 = p - q y we have

But, for us, this is an aside. The point is that onl?- in t,he exceptional cases just described the square root of a polynomial has a continlied fraction expansion with collvergents growing in height at a reasonable exponent,ial rate. One guesses t>hat in all but thosc cxceptional ca.ses - in which one expects some sort of quasi-periodicity for the continued fraction expansion of all generators of the quadratic cxtmsio~i field - t,he cont,inued fraction expansion of an algebraic power series will be 'generic' in t,he spirit suggested in the ~a r l i e r sections of our remarks.

L$'it,h a finitc basc field all these expansions will be periodic. However. only in the except,ional cascs (wlien there is an appropriate t,orsion divisor on the .Jacobian) should we expect that t,he cont,inued fraction expansions have good reduction allnost everywhcrc. It may be interest,ing in t,he generic case t,o see just how the length and complexity of the period varics as one considers rtductions rnodulo p with diffcrcnt primes p .

That mattcr is not totally unexplored. One reason to study the coritirlued fraction expansion of is to lcarn about the expansion of for intcgcrs rr . This is the theme uriderlj-ing such studies as [7]. hloreover. it is a theorem of Scliinzel [18] that the lrngt,h of the pcriod of tht, c:ontinucd fraction expansion of JD(n) is bounded as n varies if and only if y = JD(I) provitles a nont,rivial ~ m i t p ( s ) - q( . c )y which moreover has p arid q with coefficient in $ ~ [ s ] ; but that last condit,ion is obtainable in effect by restricting n to certain arithmetic progressioris.

The issue of 'convergentms' with coefficients of reasonable height is relevant in the theory of diophantine approximation. Recent work of Bornbieri and Paula Cohen [4] shows that t,he phenomena briefly described above have their a d o g u e in t,he simultaneous Pad6 approximation of algr,braic: fur~ctions of dt,grt:c, highrr t ha11 two.

5. Acknowledgements

hly rernarks linashamedly constitute a 'confcwnce paper' to the extent that they describe work and thinking in progress and that, in their first three sections. the) have subst ant ial intcrscction with earlier such remarks [14]. That intersection. however, seemed essential both to allow this work to be read indcpmdcnt 1) and to

REDUCTION O F CONTINUED FRACTIONS O F FORMAL POWER SERIES 355

allow irit, seaniltxly to int erweave explanations of phe~iomena that had astonislied me a !-car ago. 1 am irldeht,ctl to rny student Xllall Chuong Trail for encoilraging mr~ to include his dramatic example of the square root of a class of polynomials with period of length 22. The presmt work is partly supported by a grant from tlic Australian Resmrch Council.

References

[1] Willianl U'. Adarr~s and hIichacl J . Razar. 'Multiples of points on elliptic curves an(1 continuc(l fractions', Proc. Sondon Alnth. Soc. 41 (1980). 481 398.

[2] .J. P. Allourhe, XI. hIcn(l&s France and A. J . van der Poorten, 'An infinite product with tmurldctl partial qnotients', Actn Amth,. 59 (1091), 171 182.

[:3] T. G. Berry. 'On pcrio(licity of conti~iucd fractions in hyperelliptic functiou fields', Arch. Math. 55 (1990), 259 266.

[4] Enrico Bornbicri am1 Paula B. Cohcn. 'Sicgcl's Lemma, Pad6 Approxinlations and Jacobians' (with an appendix t ~ y Uni1)crto Zannicr); to appear in the De Giorgi volurnc. Anrlali Scuola Norr~lalc Superiorc. Pisa

[5] David G. Cantor, 'Oil thc corrtiiluetl fractions of quadratic surds', Actn Arzth. 68 (199-1), 2%; ,305.

[ f i ] LIicl~cl Dckkiug. XIichcl hIen(i&h France arid Alf van dcr Poorten. 'FOLDS!', The Alathe- rnat7rid I~i,te11z,qer~t:t~~. 4 (1082). 130-138; 11: LSyrrnri~try disturbed'. zbzd. 173-181; 111: 'More morphisnrs', ah7d. 190-195.

[7] E. Duhois ct R . Paysan-LC Roux. 'Stir la longeur tiu dcvelopperricnt erl fraction continue de m'. A S ~ ~ ~ Z S ~ U C 198 200 (1991), 107 119

[8] Evcrctt U'. Howe, Frar~ck Leprevost. and Bjorn Poonen. 'Large torsion subgroups of split Jacobiarls of curvcs of gcrrus two or three'. preprint .57, September 8 , 1998. located at < http://~x~ww.~riatl~.~~iuc,cdu/Algebraic-Nmber-Theory/ > .

[I)] Donald E. Kr~u th , T ~ L C Art of Computer Programming, 2 Se~nirlurncrical Algorithms (1969: 211tl printing). pp:HiOff'.

[10] B. hlazur, .hIodular curves and the Eiseristcin ideal', Inst. Hnutes ~ t u d e s Scz. Publ. Alnth. 47 (1977). :33 186.

[ I l l hlichel hIetltl?s Fra~icc, 'Sur les fractions contirlucs limitdcs'. Acta Arzth. 23 (1973). 207 21.5. [12] R. A. hlollin. A. J . van tler Poortcn an(l H. C. \.lTillianls. 'Halfway t o a solution of S 2 - IIY' =

. ' ( ' . .I. Thc:or.~c d c Norr~bri's de Bordei~u.~ , 6 (10144). 421 -159 [13] A. J . van tlcr Poortcn, 'An iritrotluctiori to continued frartions'. in J . H. Loxton and A. .J.

van der Poorten cds.. Dzophant7r~e Annly~7s (Cambridge University Press, 1986). 149 138. [14] 41f van tlcr Poortcn. 'Formal power series and their cor~tinucd fraction rxpansion', iri Joe

Buhlcr erl.. Algor7thmzc N u r ~ i h ~ r l'hcory (Proc. Third Intcrnatiorial Syrnposium, ANTS-111, Portlan(1. Ort:gon, .June 1998), Springer. Lecture Notes in Cornputer Science 1423 (1998), 12pp.

[l>] A. .J. van tler Poortcrl am1 ,J. Sliallit, 'Foltied co11tirnlt:d fractions'. J . Number Theory 40 (1992), 237 250 .

[l6] A. . T . van (lcr Poorten a i d J. Shallit. 'A specialiscti contir~uctl fraction', Cnnad. J . Afath. 45 (1!)93), 10fi7-1079

[17] C:. N. Raney. 'On continucd fr;~ctions and finite automata ' . Math. Ann. 206 (1973). 265 283. [18] A. Schinzcl. 'On some prot~lerrls of the arithmetical theory of continued fractions II', Acta

A r ~ t h . 7 (1962). 287 298.

Conterrlporary Mathematics \.'c,lurnr 236, 1999

Some Observations in Frequency Analysis

Haakon Waadeland

Ass~~ac . r . Recently there has been some interest in the use of Szego polynomials (or morc generally orthogonal rational functions) as a tool for solving the frequency analysis problem for trigonometric signals. One step in the method is to create. from a measure defined by observations of the signal. a family of positive definite sequences of moments. This is, in different ways, used to construct a family of Szego polynomials. Asymptotic properties of these lead to the unknown frequences. In the limit processes involved in the method the sequence of asymptotic values of the moments is no longer positive definite. Some Toeplitz determinants, who else could have been used to compute the limits of Szego polynornials, are zero, and hence useless for that purpose. This has, so far, been dealt with in essentially two different ways: by going to subsequences or t o introduce an additional parameter. In the present paper a quite different approach is presented in two versions. Both start with the "useless" determinants. One version is t o perturb one of the moments, the other one is t o rcplacc one (or more) of the clenlents by zero. Under some mild comli- tions both versions lead to polynomials where some of the zeros immediately produce the frequenccs.

The method was a t first merely an observation of a strange (surprising) property, hut was after a while properly established.

Dedicated to L. J. Lange on the occasion of his 70th birthday

1. Background

A signal is received, assumed to be of the form

where t denotes time (seconds). f j = - f P J for j = 1 , 2 , . . . ,I. fo = 0. A_, = AJ E R. The numbers f , are the frequenczes, the numbers IA,I the nmplztudes. The frequency analyszs problem is to determine the unknown frequencies from a set of observations, made at times m . At. where At is chosen such that we have reason to believe that 27r f,At < T for all frequencies. Now, let for all J

1991 Mathematacs Subject ClasszJ?cataon. Primary 42605; Secondary 94Al l Supported by The University of hfissouri.

358 HAAKON WAADELAND

and let the observed sigiial value at thc time r r ~ . At bc denoted s\ ( m ) . The observatioiis are then givrri by

a i d 0 elsewllcrc. IZ'e dssunie that s\ (0) # 0. The ilunibers u/, are sonietlnies called normalized frequencies. Ile shall here call then1 frequencics, and the question of cletcr~niliiiig them shall be referred to as the frequency analysis problem.

A method for solving the problern (with roots back to \$'iener and Levinson) may br roughly described in the following way: Froin t,he signal values a cert,ain absoIut,clj. coritiriuous rncasure on the unit circle is constructed. This givt~s rise t,o ail inner product, and in turn to rnoinerlt,s and nionic orthogonal polynonlials (Szrgo polynomials) on the unit circle. Asymptotic values of some zeros of the polynomials then lead t,o the frequencies. Recently ort,hogonal polynorn,ia,Ls have been replaced by orthogonal rattonal fun,ctions.

It is beyo~ld t,he scope ant1 the purpose of the prcscnt paper to describe this method. or rather these methods. since t,here arc variat,ions, essentially obtained by different choiccs of the measure. Instead we refer to t.he papers [I]. [ 2 ] . [3], [4], [5 ] and. for thc different choiecs of measures. thc paper 171.

Tlw method depends heavily upon tlw possibility of finding the Szego polynomials (or more generally t h t orthogonal rational functions). The best way, from a practical point of vicw, is to use thc Levinson algorit,hin. Another way uses continued fract,ioris. Prrron-Carath4odory fractioris for Szegii-polynomials, Nevanlinna- Pick fractions for orthogonal rational functions. A t,hird w q , although not a vcry practical one. is the one of strongest relevance to the present paper. There (in the polynomial case) t,he Szegii polynomials arc given as det,errninant formulas. wherc tlie elements are irlornerits. If, for a fixed and a fixed d e ~ r e e n the Szego polynoiiiial is denoted by p ! , \ ) ( z ) . thc formula is

where D(,' ' is the value of the upper left corner n x n Torplitz- determinant

\Ye bllall not be mterested in the explicit expressions for p i L ) , but we need to kiiow tlie following rclation (see v.g. 121):

111 thc followiiig. p,,, shall always mean thcsc, abyrnptotic values of moments.

1%. dcfiiic r i l l := 2 1 + L. where L = 1 if A. # 0 and 710 := 21 if Au = 0 (i.e. r , , , is t 1 1 ~ 1111irrber of frrcp(m~ies). A c r u ictl point in the solution of the frcquericy

SOME OBSERVATIONS IN FREQUENCY ANALYSIS 359

analysis problem is that the following holds:

i m ( z ) = (z - 1 ) ( z - e l " l ) ( z - P'"J A - x 1.

J = 1

Normally the number of frequencies is not known. But for any fixed rL the sequence

{&,") is uniformly bouridcd on the closed unit disk. An evcn more important point is that for fixed n > no an) convergent subsequence has a limit polynomial with (1.6) as a factor.

i%'e shall for convenience introduce x, = cos~j , . which leads to the equality

(1.7) ( Z - e-'"'l)(z - e7" '~ ) = zL - 2x3z+ 1.

We also have cos mij, = Tm(xJ), where T,, ( z j ) is the Tchebycheff polynomial of the first kind and degree m. Having introduced this, the formula (1.5) for the moment limits may be written in the form

For any fixed N the sequence { p ! , ~ ) / l ~ ) , , is positive-definite, and hence the denominator determinant in (1.4) is always positive, and the forrnula produces a sequence of Szego polynomials for every fixed N. This changes if we let N tend to infinity. The sequence {p,,,} is exactly no-definite, implying. in view of (1.4). (1.5) and (1.6). that for n = no we have

I Z . . . Zrf I whereds for n-values greater than no, we get a fraction where numerator and denominator both are 0 (see e.g. [ 2 ] ) . There are different ways of deahng with this: To go back to (1.4) and go to subsequences, or to introduce an additional parameter etc.. all described in the papers referred to in the present paper. Here we shall look at the numerator determinant in (1.8) for the case rL > no. but w ~ t h some changes in the elements. These changes lead to certain observations, which at first sight may look surprising. In the next section these. observations will be described, and some examples will be presented. In the final section a key theorem will be stated and proved. By using this theorem the observdtions may all be explained (proved).


2. The observations

If we expand this in powers of t , the "constant term" (non-epsilon term) will vanish, and we get

(2.2) 6 . Q b ) + 0 ( f 2 ) ,

where Q(z) is a polynomial in z, possibly identically 0.

Moment perturbat~on Take in the numerator determinant in (1.8) n = no + 1 and replace for a fixed

rn the moment prr, by p,,, + F . In the particular case where nl = 0 this looks like

Observation. For all J we have

(2.3) Q(e"1) = 0.

If Q ( z ) is not identically zero (which seems to be the "usual" case), it is a polynomial of degree no + 1 if m < no + 1 and of degree no if rn = no + 1. in both cases with the polynomial in (1.6) as a factor.

(2.1)

Example 1. We have a signal with four distinct frequencies &dl and *w2, 0 < lJJJ < 7r.

Take the amplitudes to be such that 2Af = 1 and 2A: = a for some a > 0. With coswl = x and c o s i ~ ~ = y the moment limits are in this case given by

Prn = Trr1 ( x ) + aTITf (y).

With n = no + 1 = 5 and the modification on we find. using hlACSYLlA:

Q(z) = 32aL(1 - x L ) ( l - y2) (y - x)' x

PO+€ PI . . . Pn PI PO+t . . . P11- 1

k - 1 - 2 . . . P 1

1 z . . . zrl

with the zeros e""-" and e*'"~, which are the ones we want (inter~sting zeros). and an additional (uninteresting) zero.

If, instead of n = no + 1 we take n = no + k , the expansion in powers of t will start with t'. i.e. it will be of the form

eh&(z) + 0 ( t h + l ) ,

where Q(z) is a polynornlal with zeros in all frequency points. or the determinant is identically zero. This possibility occurs for instance if the c-modification is made on p,,,, where m 2 no + 2. since in that case the highest possible power of r in the exparisiorl is 5 k . The last remark came out of a mathematical discussiori with Vigdis Petersen.


Example 2. Take a signal with two frequencies *w, 0 < w < T , and with amplitude I/&'. Take n = 4 = no + 2 and let the 6- perturbation be on p2. Then k = 2. By using MACSYMA we find

with the interesting zeros eS7"' and the uninteresting zeros

for z # 1 1 4 . For x = I/&' there is one uninteresting zero. z = 0

Observe (here and later) that the uninteresting zeros (as well as the interesting ones) are independent of the amplitudes.

A remark on the uninteresting zeros: In the "established" ways in frequency analysis the uninteresting zeros are all in the closed unit disk, the reason being that they are limits of zeros for Szego polynon~ials, whose zeros are all in the open unit disk. Here we have no reason to expect the uninteresting zero to be in the disk, since the polynomials we get by the €-perturbation are not Szego-polynomials.

Moment "killing". Again we take n = no + 1 in the numerator determinant of (1.18).

Observation. Let 1 5 p < no + 1 and 1 < q 5 no + 2. Remove row number p and column number q from the determinant. The remaining determinant is then either identically equal to 0 or it is a polynomial of degree n,() or no + 1 with

as a factor.

An alternative form of the observation follows from the expansion of the determinant according to row number p. Let at, be the element in row number 1 and -

column number m, and let

(2.4)

Al,, denote the cofactor. The determinant is

for n = n,o + 1, which is identically 0. Hence

The left-hand expression is the determinant (2.4) with the element a,>, replaced by 0. The right-hand side is, apart from the factor a,, and possibly the sign, the determinant in the observation as stated above. Hence. in the observation we may replac~ the determinant where row number p and colurnn number q are removed,


by the original detr~iniliant. but where the element in the pth row and qth col~imn is replaced by 0 as seen below:

The removed element is p,,, for some rn. If this element happens to be 0. its removal will not change the value of the determinant (2.4). and we will still have a determinant identically equal to 0. That may (exceptionally) happen also in other cases.

Example 3 h r a signal with the five frequencies. 0 arid f dL, k = 1 ,2 and the arnplitudcs

A(, = 1, 2'44 = a. 2A? = b the inoinent limits arc

p,,, = 1 + aT,,, ( s ) + bT,,, (y). a > 0. 6 > 0,

w l i e r e s : = c o s d l , ~ : = C O S ~ / ~ . Replace thc element in row number 2 and colurnn nurnber 4 by 0. This leads

to the following polynomial of degree 6 in z. factorized by I\IACSYhlA:

In addition to t h t zeros z = 1 and z = c*'-\k = 1,2, associated to the frequencies, we also have the uninteresting zero at z = - 1. Exceptional frcqucncies. urherc t h ~ process leads to 0. may be found from the factor without z.

Let it briefly be mentioned, that for n = n o + k , where k > 2, replacing k elements, all in different rows and columns. may lead to related observations. 1%. re5trict the discussion to one example.

Example 4 Take d signal with two frequencies &UJ. take n = 5 and replace the eleincnts in

the places (5.2). (2.3), (3,s) by 0. This leads to a polynomial of degree 5, computed and factorized by hIACSYhlA to be

(z' - 2sz + 1)(2.zz ' + 4sLz' + 8x3z - 21.2 + 4s' - 1).

The fir5t z-factor has the two interesting zt3ros, the second one the thrcc unintcr- (,sting ones. Here the d-values 7r/2,7r/3 and 2 ~ / 3 are exceptional values. for which the cleterlninant with the three elements replaced b) 0 is identically cqual to 0.


3. The key theorem

All observatiorls described above may be proved by using one particular key theorem. the one justifying mathenlatically the observation in the no + 1-case of what may happen if one clement is replaced by 0.

THEOREM 1. Let w,, 1 < J < I be dzstznct real numbers , all zn t h e z n t ~ r u a l (0. T ) . and a,, 1 5 .I 5 I be arbztrary posztzve numbers . Wz th z, := cos J, me d ~ f i n e , for m = 0,1,2, . . .

I

(3.1) ~ 7 7 ~ = aJTm(x3). 3=1

P'or no := 21 let P(z ) be t h e value of t h e (no + 2) x (no + 2) - determznant

where the p,,, in place (p, q) i s replaced by 0: 1 < p < no + 1, 1 5 q < no + 2. T h e n , if P(z) is n o t identically equal t o 0, i t i s a polynomial of degree no + 1 o r no, divisible by

Since here no = 21 we have in the signal the (inessential) restrict,ion A. = 0. We shall return to this later.

Proof, part 2 . In part 1 a lemma will be proved, permitting us to replace. without changing the value of the determinant. the two rightmost columns by two new columns of a special type. In these columns all elernents are 0, except possibly for the ones in row number p and in the bottom row. The two new elements in row nurriber p are constants depending upon the other elements in the same row, and the two bottom ones are the polynomials (3.3) and z 1nultip1lc.d by (3.3) ( n o = 4, p = 4 in the illustration):

Expanding the determinant according to the last column we find:


This shows that P ( z ) will be of the form

I

(3.4) P ( z ) = ( C z + D) n ( z d - 2zjz + 1). 3=1

If C = D = 0 the determinant P ( z ) is identically equal to 0. This will for instance be the case when the element to be replaced by 0 is already 0, but also in other cases (for special values of the frequencies). If C = 0, D f 0 the determinant P ( z ) is a polynomial of degree n 0 = 21, with the frequency points ~ " W J as its zeros. This will for instance be the case for q = no + 2, in particular in the trivial case (p, q) = (no + 1. no + 2). For C # 0 we have a polynomial of degree no + 1, with the frequency points as its zeros and an additional zero at z = - D / C . If in particular D = 0, the additional zero is z = 0. This happens if q = 1. Observe the trivial case

(P. q) = (131).

Proof, part 1. Since the elements are linear combinations of Tchebycheff polynomials, a natural approach to a simplification of the determinant is to use the recurrence relations for these. Repeated use will lead to a step-by-step removal of the variables xJ. This method is closely related to the method used by Vigdis Petersen in [6].

It is convenient in the proof to use also the negative subscripts for the moments (and rnornent limits). although we have avoided it in the formulas so far, since p-,, = plrl. The moment limits are then given by

Observe also, t,hat the following holds:

For nL 2 2 and for m 5 0 this is trivially true. For m = 1 we see it by inspection, since Tl ( x ) = x and (x) = 1.

LEMMA 2. Let n l be an znteger, 0 5 m < no + 1. Then there exzsts a set of real numbers cl . ~ 2 , . . . . C O , zndependent of m, such that

Proof of Lemma 2. We have, for 0 5 s < no - 2


Next we repeat the process. but with x2 instead of xl and find in the same way

After I - 1 steps ( with in turn X I , x2. . . . , X I - 1 ) we get the expression

and the last step, now with XI, gives 0. With s = 0 this process yields a linear combination of pl l I , , C L ~ , ~ - ~ , . . . pr l , -nO equal to zero, and where the coefficients are independent of m, and where the coefficient of p,, is 1. Lemma 2 is thus proved.

Remarks For no = 2 the relation ( 3 . 6 ) takes the form

7 - ( 1 + x ) ~ ~ - l + (2 + ~ 1 ~ 2 ) ~ r 7 1 - 2 - 2(x1 + ~ 2 ) 1 * . r ! i - j + /1r!! -1 = 0.

It is easy to go from one value of no in ( 3 . 6 ) to the next one (difference 2 ) by the same type of recurrence as in the proof.

Lemma 2 shows that there is a linear combination of the colurrms 2 to no + 2 in ( 3 . 2 ) such that the "new'' (no + 2)th column has merely zeros except possibly in place p and in the bottom place. By applying the same linear combination to the columns I to no + 1 we accomplish the sarnc in the "new" (no + 1)th column. It remains to find out what happens to the bottom place: Since the bottom elements in the process are subject to the same procedure as the others, we find the polynomials in the bottom elrnlents of the new two rightmost elements by replacing in (3.6) the moments plrl-n by the power z""+'-' fo r the element in row no + 2 and by z""-' for the element in row no + 1. From the above remarks we see that for no = 2 we

get

z ' - 2 x , z 2 + Z, zL - 2.r12+ 1 ,


and z times the latter in row no + 2 = 6. Rather than using (3.6) we may apply the procedure drscribed above directly to { z l )!=,ll,+, . leading to more convenient forms of the new elements. After one step we get {zl (zL - 2 x p + l ) } ~ = , l , , p , . after two steps {zr (z' - 2x12 + 1)(z2 - 2x22 + l))!_,,,,pj etc., and in the end the (no + 2, no + 2)th element in the new determinant. The (no + 2. no + l ) t h elemtnt is obtamed in the same way, if we start from {z'))',,,~,. This conrludes part 1 of the proof of Theorem 1. and hence part 2 is justified. We thus have a complete proof of Theorem 1.

In the case when A. # 0 the procedure is the same, except for one additional step at the end, producing the factor ( z - 1) in the two rightinost bottom elements, and thus in (3.3) in the theorem.

Let it finally be mentioned, that in Theorem 1 the rlurnber 0 is not essential. as ma) be seen from the proof, the theorem will still be true if it is replaced by any other fixed number.

Remarks \Ve need to indicate the bridge between Theorem 1 and the observation on the

t-perturbation. Take first n = no + 1, and replace a fixed p,,, by p,, + t . Then the determinant may be written as a sum of three types of terms: First the original determinant without any 6. That determinant has the value 0. Next t times a sum of determinants obtained from the original one by removing exactly one row (not the bottom one) and one column. Finally a term 0 (c2 ) . From Theorem 1 (keeping in mind the two equivalent forms of the observation) it follows that each of the determinants where exactly one row and one column are removed either is identically 0 or divisible by (3.6). Thus the observation on the 6-perturbation is established as a mathematical fact in the case n = no + 1. That we, for n = no + k . in the t- perturbation get the factor t' follows as in [6].

If. instead of having n = no + 1 in Theorem 1 we have n = no + k , where k 2 2 , we use the same column operation as the one established in Lemma 1. but this timt not only 2 times, but k + 1 times, leading to a determinantt, where the k + 1 right~nost columns have merely the element 0, except for k distinct rows. say p l . p2, . . . , pk . where we have constants possibly different from zero, and thc bottom row. where we have the polynomial (3.6). multiplied by powers of z. in turn z", z' , . . . zL. Expansion according to the rightmost colurnn (number no + k + 1) leads to a polynomial of degree k multiplied by the factor (3.6). If the first polynomial is not identically 0 it gives (up to) k uninteresting zeros, whereas the second polynomial gives the interesting ones.

References

[I] W.B.Jones, 0.Njastati. W.J.Thron and H.Waadcland, Szego polynomaals applzed t o frequency analyszs J . Comput. Appl. hlath. 46 (1993), 217 228.

[2] W.B.Joncs, O.Nj&stad and H.Waadcland, A n alternatzve w a y of us ing Szegii polynomials zn frequency analyszs, in S.C.Cooper and W.J. Thron eds. Continued Fractions and Orthogonal Functions, Marcel Dekker, New York, 141 152, 1994.

[3] O.Njastad and H.Waadeland General Szego theory i n f r equemy analyszs, J . Math. An. and Appl. 206 (1997), 280 307

[4] O.Njastarl and H.Waadeland, Asympto t zc propertzes of zeros of orthogonal rational functzons, J . Comput. Appl. hlath. 77 (1997), 25F -275.

[5] K.Pan and E.B.Saff, Asympto t zcs for Zeros of Szego polynomaals assoczated 111zth trzgolzomct- rzc polynomzal szgnals, J . Approx. Theory 71 (1992), 2 3 9 2.51.


[6] V.Petersen. A t heorem o n Toeplztz de t e rminan t s wzth elemeizts contaznzng Tchebyche f l poly- nomza l s of t he first kind, Det Kgl . Norske Vitiensk. Selsk. Skr. 4 (1996). 1 15.

[7] V.Pctersen, O n measures zn frequency analys is , J . Cornput. Appl. Math. . To appear.

Some properties of Hermite-Pad6 approximants to e"

Franck Wielonsky

dedicated to Jerry Lunge on the occasion of his 70th birthday

ABSTRACT. We investigate questions such as convergence, differential equations, location of zeros of Hermite-PadB approximants to eZ and display some numerical experiments concerning the distribution of zeros. We consider the known results about the more elementary Pad6 approximants to e2 as a general background for the discussion.

1. Introduction

Padk approximation may be seen as one of the many ways of performing approximation t o analytic functions in the complex plane. One of the main features of the Padk approximants comes from the algebraic nature of their definition. Through- out, P k will denote the set of polynomials with complex coefficients, of degree at most k.

DEFINITION 1.1. Let f be a function analytic at the origin. The Pad6 approximant of degree (m, n) is defined as the rational function P,,,/Q,,, such that

(Qrn,, f - P m , n ) ( ~ ) = O ( Z ~ + ~ - ' ) as z -+ 0,

with P,,, E P, and Q,,, E P,.

Thus, given the Taylor's coefficients of the function f at the origin. the Pad6 approximants can be explicitely computed by solving a set of linear equations. There exists a whole theory based on algebraic tools such as determinants which leads to numerous identities, recursion relations and algorithms. In this connection, one can also mention the strong links that exist between Pad6 approximants, continued fractions and orthogonal polynomials.

The other aspect of the theory is the analytic aspect and the main interest, here, lies in properties such as convergence, asymptotics and distribution of zeros. In this respect, the Padk (or Baker-Gammel-Wills) conjecture plays a prominent role which predicts that , given a meromorphic function f . there exists an infinite subsequence N C N such that the Pad6 approximants of degree (n. n), n E N converge locally uniformly to f , away from the poles of f . as n tends to infinity (cf. [Stag71 for a recent overview of this conjecture). In general, analyticity is not

1991 Mathematzcs Subject Classzficatzon. 30E10. 30C15, 41A21

370 FRANCK WIELONSKY

sufficient in order to ensure convergence of the full sequence of Pad6 approximants, as the existence of spurious poles shows (cf. [Lub92, Per57, Wal741).

In this paper, we shall describe a few results and numerical experiments that concern the analytic aspect of a classical generalization of Pad6 approximants, namely the Hermiteepad6 approximants.

DEFINITION 1.2. Let i f o , . . . , fm) be a vector of m + 1 functions analytic at the origin. For any multi-index n = (no , . . . , nm) E Rim+', the (latin or type I) Hermite-Pad6 approximants of degree n are defined as the nonzero vector of polynomials

Let us just recall that there exists another type of Hermitepad6 approximants. the german type or type 11, which consists of simultaneous rational approximants. Concerning the algebraic and analytic aspects of Hermite-Padi. approximants, we refer to [BGM96, dB85, Coa66, Coa67, Mah681 and [AS92, Nut84, Stat381 respectively.

In the sequel, we shall only consider the approximants in Definition 1.2 when specializing the choice of t:ie vector of functions ( f o . . . , f7n) to be the vector of exponentials (1, eZ, . . . . em '). An important property of such a vector is that it constitutes an example of a perfect system. It means that for any multi-index n = (no , . . . , n,). any solution Ao, . . . ,Am of (1.1) satisfies

Hence, the solution to (1.1) is actually unique, up to a constant factor. In the subsequent sections, we shall start from the known results about Padk

approximants and discuss questions such as convergence, differential equations, location and asymptotic distribution of zeros of Hermite-Pad6 approximants.

Let us terminate this introduction by mentionning the well-known application of Padk approximation to number theory, which was initiated by Hermite, in proving the transcendence of e. A few references here, among others, are [BeuBl, Pr696, Ass98, dP791.

2. Convergence of Hermite-Pad6 approximants to eZ

Let us first take a look at the Pad6 case of type (m, n). We thus consider two polynomials P,,, and Q,,,, of respective degree m and n , such that

Note that upon dividing the previous equation by eZ and changing z into -z, we get

Pm,n(-z)eZ - Qm.n(-z) = O(z m+n+l 1.

which implies, by uniqueness, that Q,,,(r) arid P,,,(-r) are equal, up to a constant. The following theorem of Pad6 shows that for a rational function, interpolating the exponential function at zero with an order as high as possible suffices to imply uniform convergence in the complex plane to this exponential function.

T H E O R E M 2.1. W i t h P,,, and Q,., satisfying ( 2 . 1 ) , we have

locally unzformly in @ as m + n 4 m. Moreover, if mln --t A, one has separated convergence, namely

In particular, i n the diagonal case m = n + m, we haue

The proof relies on the integral expressions of P,,, and Q,,,

Pm,n ( z ) = 1 / e- ' ( t + a)"'tndt,

( n + m)!

Explicit forms are given by

(cf. [Per57]). Let us proceed with Hermiteepad4 approximants, by considering the vector of exponentials ( 1 , eZ, . . . , e m Z ) . This is one of the few cases where integral expressions can be given for the solutions to ( 1 . 1 ) . Indeed, it is easily checked that the formulas

where Co is a circle centered at the origin and of radius less than 1: define polynomials of degree np - 1 satisfying (1 .1 ) . Then, it is natural to ask whether the previous theorem can be generalized to Hermiteepad6 approximants. The following result, whose proof can be found in [Wie97]> answers this question in the diagonal case, that is when considering m + 1 polynomials Ao, . . . , A , such that

and all the polynorriials Ao, . . . A, are of degree less than the same constant integer n E W .


T H E O R E M 2.2. Let Ao, . . . ,A , be the Hermite-Pad6 approximants t o the exponential funct ion given by ( 2 . 4 ) ) with nl = n, 0 < 1 < m, of degree less t h a n n.

m-l be the set of the m critical points, that i s the rn roots Let{-p-rl, ,O < 77p < l),=O of the derivative, of the Pochhammer polynomial

( z ) , = Z ( Z + 1 ) . . . ( z + m - 1).

T h e n , there exist some explicitly computable nonzero constants P , , ~ , 0 5 p < m / 2 , such that , as n -+ c o ,

Consequently, for n large, one can define & as the polynomial obtained upon dividing A, by i t s nonzero constant coef icient . T h e n

( 2 ) locally uni formly in 6. If m i s even, let A": (resp. A,/,) be the subsequence of

polynomials Am12 corresponding to even ( r e s p odd) indices n. T h e n , A ( L ) ~ i s a n

odd polynomial and A": a n even polynomial. Moreover, there exists a n explicitly computable nonzero constant pm/2,n such that , as n -+ co ,

For n large, let A s 2 and A"2 be the polynomials obtained upon dividing ~ ( 1 : ~ and

A ( ~ ) respectively by the nonzero derivative a t zero and nonzero constant coef icient . m/2

T h e n , as n + c o ,

uni formly o n compact subsets of @.

The proof relies on applying the saddle point method to the integral expressions ( 2 . 4 ) of the polynomials A,. Using in the same way, the integral representation of the remainder term R,

where C, is a circle centered at the origin and of radius greater than m, one may also show that in the diagonal case

uniformly on compact subsets of @: as n -+ co. The derivation of all the previous asymptotics for the non diagonal case can be obtained similarly.

R E M A R K 2.3. From Theorem 2.2, one easily recovers the assertions of Theorem 2.1 in the diagonal case. Indeed, the unique critical point of z ( z + 1 ) is -112 so that 770 = 112.

EXAMPLE 2.4. When m=2 and

Ao(z) + A1 (z)ez + ~ ~ ( z ) e ~ ' = o ( z ~ ~ - ' ) ,

we consider the critical points of z(z+ l ) (z+2) which are -1 + l / f i and -1 - 1/& so that

70 = 1 - ~ / , h , r ) ~ = l/,h.

Then, from Theorem 2.2, one gets that

A2(z) (- l )"- l~o,ne -(I-l/&)z ,

n where PO,, may be seen to equal 1 3&

3 4 . G (7)

3. Some differential equations

Let us now establish the differential equations satisfied by the Hermiteepad6 approximants Ao, . . . , A, such that (2.5) holds. For clarity, as before, we shall limit ourselves to the diagonal case, though the general case can be treated in a similar way. First, in connection with the Pad6 approximants Pn := P,,, and Q, := Q,,, defined by (2.1), with m = n, we set

~ " ( 2 ) = e-z/2z-n Pn(.).

Then, w,(z) satisfies Whittaker's equation (cf. [Olv54, p.2601)

or, equivalently, Pn satisfies

(3.1) nPn(z) = (z + 2n)PA(z) - zP l (z ) .

Also, from the remark after (2.1), we deduce that

(3.2) nQ, (2) = (z - 2n)QL(z) + zQz(z).

Let us now derive the generalization of (3.1) and (3.2) corresponding to the approximants Ao, . . . ,A,. Consider the contour integral (2.7) in the diagonal case, that is, n1 = n, 0 < 1 5 m and note that for any polynomial G, we have

where D denotes the differential operator dldz. Set

(3.4) L(t) = t(t - 1 ) . . . (t - m).

Then we can apply partial integration to (3.3) with (n - l )L1 instead of G and obtain

- - On the other hand, the right hand side of the previous equation equals zL(D)R(z). Hence we get the differential equation


Since the functions 1, e Z , e2", . . . , em" are linearly independent over the rational functions, the differential equation holds for each of the summands Ap(z )epz of R ( z ) . Hence

z L ( D ) ( A p e p Z ) - (n - l ) L 1 ( D ) ( A , e p z ) = 0 .

Because of the identity D(epZu) = epz(D + p)u , this implies that

We summarize the result in the next theorem.

THEOREM 3.1. Let Ao, . . . ,Am be the diagonal HermiteePade' approximants to the exponential function satisfying (2.5), of degree less than n, and let L be the polynomial defined by (3.4). Then, the following differential equations of order m + 1 are satisfied:

EXAMPLE 3.2. Let

define, up to a constant, the Hermite-Pad6 approximants of degree 4 of the vector (1, e Z , ezZ , e3'). Assuming n = 5, m = 3 in Theorem 3.1, it is straightforward to check that

REMARK 3.3. The differential equation (3.5) relates A$,+') and the rn + 1

polynomials A,, A P ) , . . . , A L ~ ) . By differentiating (3.5) several times, we get a

linear relation with polynomial coefficients between any derivative A!). j 2 m + 1 and the polynomials A,, A!), . . . , A $ ~ ) . On the other hand, from the analog of formula (3.3) for the polynomial Ap,,, where, here, the second subscript denote the degree, we deduce that

These observations allows one to compute, for each p, 0 < p < m. a recurrence relation involving the polynomials A ,,,, . . . , A, ,,-,- '. Indeed. from what precedes,

( 1 ) there are linear relations between Ap,n-J and Ap,,, Ap ,n , . . . , A$;). Hence, a linear relation between A ,,,. . . . , A ,,,-,- 1 can be established.

4. On t h e zeros of Hermite-PadB approximants to eZ

In this section, we shall review some known facts about the zeros of Pad6 approximants to eZ and display some numerical experiments concerning the zeros of Hermite-Pad6 approximants. The first result in studying the zeros of such approx- inlants may be the article of Szego [Sze24], which considers the zeros of the partial sums s,(z) = z k / k ! of the Taylor expansion of e z . Note that s,(z) is the Pad6 approximant of ez of degree (n , 0 ) . Szego showed that the normalized partial

sum sn(nz) has all its zeros in Izl < 1 for every n > 1, and that i is a limit point of zeros of {~ , (nz ) )T=~ iff

= 1 and 121 < 1.

The so-called Szegb' curve determined by the previous equations is shown in Figure 1. Concerning general Pad6 approximants, Saff and Varga have given in a series of papers (cf. [SV75, SV76, SV77, SV78] and the references therein) numerous results concerning the location of their zeros. Mainly using the three-term recurrence relation (or Frobenius relation) and the second-order differential equation satisfied by these approximants, they could prove the existence of a sector, alternatively a parabolic region determined by the type of the approximants, free of zeros. A sharp lower bound as well as an upper bound on the modulus of these zeros could also be established in this way. Moreover, by means of the saddle point method applied to the integral representation (2.2) and (2.3), asymptotic estimates were obtained, from which the asymptotic distribution of the zeros of the normalized Pad6 approximants and of the error function could be determined. The eye-shaped curve which consists in the limit points of zeros, poles or zeros of the remainders generalizes the Szego curve. We refer the reader to the original papers for complete statements of the theorems and to jBGM96, pp.268-2741 for a nice summary of these results. Let us now proceed with Hermiteepad6 approximants of the vector (1, eZ, . . . ,em'). To the author's knowledge, such precise results as above are not yet available for the zeros of these approximants. We only state the seemingly weak upper bound (cf. [Wie97]):

PROPO~ITION 4.1. For any m > 1 and n > 2, all the zeros of the Hermite-Pad4 approximants A, (z) satisfying (2.5) lie in

FRANCK WIELONSKY

where it is understood, in case p = 0 or p = m, that the sum in (4.1) ranging from k = 1 to 0 vanishes.

Finally, we give some numerical results about these zeros. In Figure 2, we have graphed the zeros of the five polynomials Ao, Al, A2, AY, A4, a11 of degree 50, such that

The 50 zeros of the polynomials A. to A4 appear in 5 sequences from the left to the right of the figure. The zeros of Ao, A2, Aq are denoted by circles "on. Those of Al and Ag are denoted by cross "+". In the two subsequent figures, Figures 3 and 4, we still represent, in the same way as in Figure 2, the zeros of 5 polynomials such that the expansion (4.2) has maximal vanishing at zero, but now, we consider non diagonal approximation. Indeed, we

respectively. We may conjecture that, with a convenient normalization, the zeros of the

Hermite-Padk approximants cluster, as the degrees tend to infinity, to fixed curves, analog of the Szego or eye-shaped curves. As in the Padk case, these curves would be determined by the different ratios of the degrees of the approximants, as they tend to infinity. It seems possible that using the differential equations in Theorem 3.1, one can obtain some information on the location and asymptotic distribution of the zeros of the Hermite-Padk approximants to a vector of exponentials. Acknowledgements The author would like to thank the referee for his helpful

FRANCK WIELONSKY

comments and especially for having supplied us with a proof of Theorem 3.1, simpler than that given in a first version of this paper.

References

[AS921 A.I. Aptekarev and H. Stahl. A s y m p t o t i c s of Hermite-Pad6 polynomials , Progress in Approximation Theory (A.A. Gonchar and E.B. Saff, eds.), Springer Verlag, 1992. pp. 127-167.

[Ass981 W . Van Assche, A p p r o x i m a t i o n theory and analyt ic n u m b e r theory , Special Functions and Differential Equations (New Delhi) (K. Srinivasa Eao et al., ed.). Allied Publishers, 1998, pp. 336-355.

[Beu81] F. Beukers, Pad6 approxzmatzon i n n u m b e r theory , Pad6 approximation and its applications (M.G. de Bruin and H. van Rossum, e d ~ . ) , Springer Lecture Notes, vol. 888, Springer. 1981, pp. 90-99.

[BGM96] G.A. Baker and P. Graves-Morris. Pad6 approx iman t s , Cambridge University Press, 1996.

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