harmonic analysis and hypergroups

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ANDERSSONILAPIDUS (eds). Progress in Inverse Speetral Geometry JAINIRIZVI (eds). Advanees in Ring Theory Ross/ANDERSONILITVINOv/SINGHlSUNDERlWILDBERGER (eds). Harmonie Analysis and Hypergroups

Harmonie Analysis and Hypergroups

K.A. Ross J. M. Anderson G. L. Litvinov

A.1. Singh V. S. Sunder

N. J. Wildberger Editors

with the assistance 01 Alan L. Schwartz Martin E. Walter

Springer Science+Business Media, LLC

K. A. Ross Dept. of Mathematics University of Oregon Eugene, OR 97403

A. I. Singh Dept. of Mathematics University of Delhi Delhi, India

J. M. Anderson Dept. of Mathematics University College London London, UK

V. S. Sunder The Institute of Math. Sciences CIT Campus, Taramani Chennai, India

G. L. Litvinov Centre for Optimization and Mathematical Modeling Institute for Technologies Moscow, Russia

N. J. Wildberger School of Mathematics Univ. of New South Wales Sidney, NSW, Australia

Library of Congress Cataloging-in-Publication Data Hannonic Analysis and Hypergroups (International Conference on Hannonic Analysis

1995 : University of Delhi) International Conference on Hannonic Analysis / editors, K. A. Ross ... [et al.].

p. cm. -- (Trends in mathematics) Proceedings of a conference held Dec. 18-22, 1995 at the

University of Delhi. Includes bibliographical references

1. Hannonic analysis--Congresses. 11. Title III. Series.

I. Ross, Kenneth A.

QA403.155 1995 515'.2433--dc21

97-31839 CIP

ISBN 978-1-4899-0158-3 ISBN 978-0-8176-4348-5 (eBook) DOI 10.1007/978-0-8176-4348-5

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Contents

Prefaee •.....•....••. VI

De Branges Modules in H2(Ck )

Sanjeev Agrawal and Dinesh Singh

Some Methods to Find Moment Funetions on Hypergroups

1

Leonard Gallardo . . . . . . . . . . . . . . . . .. .. 13

About Some Random Fourier Series and Multipliers Theorems on Compaet Commutative Hypergroups Marc-Olivier Gebuhrer ................ 33

Disintegration of Measures Henry H elson ..... . ......... 47

Multipliers of de Branges-Rovnyak spaees II Benjamin A. Lotto and Donald Sarason ........ 51

On Hartman Uniform Distribution and Measures on Compaet Spaees R. Nair ..... . ............ 59

Hypergroups and Signed Hypergroups K enneth A. Ross . . . . . . ...... 77

Three lectures on Hypergroups: Delhi, Deeember 1995 Alan L. Schwartz . . . . . . . . . . . . . . . . . . . . . . 93

Harmonie Analysis and Functional Equations Henrik Stetkaer . . . . . . . . . . . . .

Actions of Finite Hypergroups and Examples V. S. Sunder and N. J. Wildberger ....

Positivity of Tunin Determinants for Orthogonal Polynomials

131

145

Ryszard Szwarc . . . . . . . . . . . . . . . . . . . . 165

Wavelets on Hypergroups K. Trimeche . . . . . .

Semigroups of Positive Definite Functions and Related Topics

183

Martin E. Walter . . . . . . . . . . . . . . . . . . . 215

Characters, Bi-Modules and Representations in Lie Group Harmonie Analysis N. J. Wildberger . . . . . . . . . . . .

A Limit Theorem on a Family of Infinite Joins of Hypergroups Hansmartin Zeuner . . . . . .

227

243

Preface

Under the guidance and inspiration of Dr. Ajit Iqbal Singh, an International Conference on Harmonie Analysis took place at the University of Delhi, India, from December 18 to 22, 1995. Twenty-one distinguished mathematicians from around the world, as weIl as many from India, participated in this successful and stimulating conference.

An underlying theme of the conference was hypergroups, the theory of wh ich has developed and been found useful in fields as diverse as special functions, differential equations, probability theory, representation theory, measure theory, Hopf algebras and quantum groups. Some other areas of emphasis that emerged were harmonie analysis of analytic functions, ergo die theory and wavelets.

This book includes most of the proceedings of this conference. I chaired the Editorial Board for this publication; the other members were J. M. Anderson (University College London), G. L. Litvinov (Centre for Optimization and Mathematical Modeling, Institute for New Technologies, Moscow), Mrs. A. I. Singh (University ofDelhi, India), V. S. Sunder (Institute of Mathematical Sciences, C.LT., Madras, India), and N. J. Wildberger (University of New South Wales, Australia). I appreciate all the help provided by these editors as weIl as the help and cooperation of Our authors and referees of their papers. I especially appreciate technicial assistance and advice from Alan L. Schwartz (University of Missouri - St. Louis, USA) and Martin E. Walter (University of Colorado, USA). Finally, I thank Our editor, Ann Kostant, for her help and encouragement during this project.

Very special thanks go to Dr. Ajit Iqbal Singh for organizing this conference. This was her brainchild and it would not have taken place without her determination and persistence. She was ably assisted by many colleagues, especially Dinesh Singh and B. S. Yadav, and by several cheerful and helpful students. Around 70 different people served on one or more of the ten committees responsible for this conference. For generous financial assistance, we are grateful to the (Delhi) University Grants Commission, Council of Scientific and· Industrial Research, Department of Science and Technology, Indian National Science Academy, and the National Board of Higher Mathematies. We are thankful to the Department of Science and Technology for a special grant and for publishing the conference brochure.

K. A. Ross

De Branges Modules in H 2(Ck )

Sanjeev Agrawal and Dinesh Singh

1. Introduction

One of the most important results in invariant subspace theory is the famous "Beurling's Theorem" [1], characterizing the invariant subspaces of the shift operator S (i.e. multiplication by the co ordinate function z) on the Hardy space H2(T).

Beurling's Theorem. Let M be an invariant subspace 01 S in H 2 (T). Then there exists an inner lunction q(z) in HOO(T) (that is, {q(z)zn} is an orthonormal set) such that

Recently, L. de Branges [3] has generalized this theorem, generalizing, at the same time, the vector-valued version of Beurling's Theorem due to Lax and Halmos [4,5,7]. L. de Branges' theorem (scalar version) characterizes the dass of Hilbert spaces that are contractively contained in H 2 (T) and on which the shift operator acts as an isometry.

De Branges' Theorem (Scalar version). Let M be a Hilbert space contractively contained in H 2 (T) such that S(M) c M and S acts as an isometry on M. Then there exists a b(z) in the unit ball 01 HOO(T) (unique up to a scalar lactor 01 unit modulus) such that

M = b(z)H2(T)

and the norm on M is given by Ilb(z)j(z)IIM = 111(z)IIH2 In [10] we have extended the theorem of de Branges (scalar version)

to the context of various Banach spaces of analytic functions on the unit disc such as the Hardy space, the Dirichlet space, the Bergman space, BMOA and VMOA. In this paper we have looked at the nature of those Hilbert spaces that are vector subspaces of the space H 2 (Ck ) and on which S acts as an isometry. Not all such spaces will look like the space

o Mathematics Subject Classification (1991 revision). Primary 47A15, 47B37. Key words and phrases. Ck-valued de Branges' Theorem, Ck-modules, shift,

Hilbert Ck-modules.

2 S. Agarwal and D. Singh

as characterized in de Branges' theorem. We show precisely which spaces will look like the space as in de Branges' theorem in terms of the norm and the vector space structure. We also give examples of spaces that, as sets, look like the space in de Branges' theorem but whose geometry is different from the type as given in his theorem.

It should be borne in mind that, unlike the result of de Branges, our results do not make any contractivity or continuity ass um pt ion between the space to be characterized and the space H 2 ( Ck ).

2. Preliminary results and definitions

C K will denote the k-dimensional unitary space. For 0: = (0:1, ... ,O:k) and ß = (ß1, ... , ßk) ECk, if we define

0:* a =: (al, ... ,ak) , o:ß (0:1ß1, ... ,O:kßk) ,

110:11% 10:11 2 + ... + IO:kl 2

then (Ck, 1I.lIk) forms a Banach *-algebra. The set {ei} will denote the standard orthonormal basis of C k where the ith coordinate in ei is 1 and all the other coordinates are zero. The basis is ordered by i. All matrices in this paper are written with respect to the basis {e;}. Let A be a nonsingular k x k matrix. For 0:, ß E C k , we define

(0:, ß) A = (Ao:, Aß) k ,

It is easy to check that (., .)A is an inner product on Ck. The corresponding norm on C k will be denoted by 11·11 A.

Lemma 2.1. 11 (., .) is an inner product on C k , then there exists a unique upper tri angular k x k matrix A whose diagonal entries are positive real numbers and such that

(.,.) = (., ·)A .

Proof. The result follows from the characterization of positive definite sesquilinear forms on Ck and the Q R factorization of matrices. For the QR factorization of matrices we refer to the following theorem in [6, pagel12]. If A is a nonsingular k x k matrix, there exists a unitary matrix Q and an upper tri angular matrix R such that A = QR. R may be chosen so that all its diagonal entries are positive and, in this event, the factors Q and Rare both unique.

De Branges Modules 3

H 2 ( C k ) will denote the space of all formal power series 2: anzn where an E Ck and 2: Ilanll~ < 00. It is weIl known that (H2(Ck ), 11.1112) is a Hilbert space, where II2:anz1112 = 2:llanll~. Further, H 2 (Ck ) is a Ck _

module under the multiplication al = a 2: anzn = 2: aanzn. Let S denote the operator of multiplication by the coordinate func

tion z on H 2 (Ck ), that is, S(j(z)) = zI(z). S is an isometry in H2(Ck ). HOO(Ck ) will denote the space of all I E H 2(Ck ) such that Ig E H 2(Ck ) for all 9 E H2 (Ck ); the product of land 9 being defined as their formal Cauchy product. Let A be a nonsingular k x k matrix. For I E 2:an zn , ifwe define IIIII~ = 2: lIanll~, then it is easy to see that (H2 (Ck ), II.II A ) is a Hilbert space, such that S(H2 (Ck )) C H2 (Ck ) and S acts as an isometry on it. •

Definition 2.2. (De Branges space) If (M, II.IIM) is a Hilbert space that is a vector subspace of H 2 (C k ), then M is called ade Branges space if there exists a b( z) E Hoo (Ck ) such that

and {anznb(z)} is an orthogonal sequence in M for every sequence {an} in Ck .

Definition 2.3. ([2]) A Hilbert space (M, 11.11) is said to be a Hilbert Ck-module if

(i) M is an algebraic (left) module over C k ,

(ii) (ax,y) = (x,ay) for every a E Ck,x,y E M, (iii) there exists a constant R such that

Ilaxll ~ R Ilallk Ilxll for every a E C k and xE M.

Remark 2.4. All Ck-modules in this paper will be Ck-submodules of H 2 ( Ck ) in the algebraic sense.

Remark 2.5.

(1) For I E H 2(Ck ), we shall write I(i) for e;j. Note that in all the coefficients of I(i), all coordinates, except possibly the ith coordinate, are zero.

(2) For aspace M ofthe type b(z)H2(Ck ), we will assume that b(i) =I- 0 for each i. Otherwise M can be considered as a subspace of H2(cr) for a suitable r < k. Thus there is no loss of generality.


Proposition 2.6. If A is a nonsingular k x k matrix, then

(a) (H2(Ck), II.IIA) is a de Branges space.

(b) If b E Hoo (Ck), then (M, 11.11 M) is a de Branges space where

M = bH2(Ck) and Ilbfll M = IlfiI A .

Proof.

(a) H 2(Ck) = eH2(Ck), where e = (1,1, ... ,1). Now

(aezm,ßezn)A = ((Aae)zm, (Aße)zn)H2

= 0 if m=l=n.

Thus (H2(Ck ), II.II A ) is a de Branges space.

(b) M is obviously a vector subspace of H2(Ck ) and S(M) c M. By part (a), (eH2 (Ck), II IIA) is ade Branges space. Thus

(aebzm, ßebzn) M

(aezm, ßezn) A

o if m =1= n.

3. The main results

We start with the converse of Proposition 2.6(b).

•

Theorem 3.1. Let (M, II.IIM ) be a Hilben space that is a vector subspace

of H 2 ( Ck) such that S (M) c M and S acts as an isometry on M.

(a) Then M is a de Branges space if and only if Ker S* is an algebraic Ck-submodule of H 2(Ck).

(b) In this case, M = b(z)H2(Ck) (where {anznb} is an orthogonal sequence in M for every sequence {an} in Ck and if b(i) =1= 0 for all (i), and there exists a nonsingular k x k upper triangular matrix A such that

Ilbfll M = Ilfil A


Remark 3.2. For k = 1, this theorem gives the scalar version of de Branges' Theorem without the contractivity condition. Later on we shall show the existence of spaces of the type b( z) H 2 (Ck ) that are not de Branges spaces.

Proof.

(a) The proof of this part of the theorem is essentially a simpler version of the proof of the main theorem in [11]. Here we prove the most important part of the proof. The missing links can be found in [11]. Note that Ker S* = M eS(M). We write N for M eS(M).We first assume that N is a Ck-submodule of H 2(Ck ). By the Halmos-Wold decomposition

M=N$S(N)$ ....

Claim 1: If bEN, then {anbzn} is an orthogonal set in M for every sequence {an} in Ck . This is easy to show and we skip the justification.

Claim 2: For each f in Ck and for all a E Ck there exist positive constants AJ , BJ such that AJ IIafl1 2 :::; IlafllM :::; BJ IlafllH2' This follows from the fact that for any f E M, the set {af : a E Ck } is a finite-dimensional subspace of H 2 ( C k ) and of M. (Thus M satisfies condition (iii) of Definition 2.3.)

Claim 3: If f E N, then fg E M for all 9 E H 2 (Ck ). Let f = 2:: anzn, 9 = 2:: ßn zn , and gn = 2::r=o ßizi. Then gn ---t 9 in H 2 ( C k )

and hence {gn} is a Cauchy sequence in H 2 (Ck ). Also,

Ilgmf - gnfll~

= IIßmzm fll~ + IIßm-1Zm-1 fll~ + ... + IIßn+1Zn+1 fll~ (by Claim 1)

= IIßmfll~ + IIßm-dll~ + ... + IIßn+1fll~ (as S is an isometry on M)

:::; BJ [IIßmfll~2 + IIßm-dll~2 + ... + IIßn+1fll~2] :::; BJ Ilfll~2 [IIßmll~ + IIßm-lll~ + ... + IIßn+1II~]

(as lIafll~2 :::; lIall~ Ilfll~2) :::; BJ IIfll~2l1gm - gnll~2'

Thus, {gnJ} is a Cauchy sequence in the Hilbert space M and hence converges to, say, h E M. Then it can be shown that [11, page 65] h = fg. This settles Claim 3.


In fact, we have shown that for f(z) E N and L:;anzn E H 2(Ck),

n

2:. ai zi f --+ [2:. an zn] f in M. i=O

Also note that we have shown that the linear span of {azn b( z) : a E Ck and n 2 O} is dense in (M, II.IIM). Now it can be shown that [11, pages 66-69] if f , gEN are such that f(i) ..1 gei), then either feil = 0 or gei) = O. Let i be such that there exists f E N such that f(i) # o. Let gEN. It is easy to see that gei) is linearly dependent on f(i). Let X = {i : there exists fE N such that feil # O}. For each i EX, choose qi E N such that eiqi # O. For i rt X, take qi = O. Let b = el ql + ... + ekqk. Then it is shown in [11] that N = {ab: a ECk}, and then M = b(z)H2(Ck). Conversely, it is obvious that M is a Ck-submodule of H 2 (Ck ).

By [ 11, page 72], N = {ab: a ECk}. This implies that N is a Ck-submodule of H 2 (Ck ). Note that the characterization obtained here is algebraic in nature. No knowledge about the norm on M is assumed.

(b) We define 11.11 on H 2(Ck ) as Ilfll = IlbfllM· As 11· 11M is a norm on M, it is easy to check that 11.11 is a norm on H 2(Ck ). Obviously S acts as an isometry in (H2 (Ck), 11.11). Also, {anzne} is an orthogonal sequence in M for every sequence {an} in C k where e = (1,1, ... ,1). Thus H 2(Ck) (= eH2(Ck),II.II) is a de Branges space. Let f = L:anzn E H 2(Ck). Let Pne --+ fe in (eH2(Ck) 11.11). Thus IIPneil --+ IIfell· As Ck is a finite-dimensional subspace of (eH2 (Ck ), 11.11), C k is a Hilbert space under 11.11. Thus there exists a nonsingular k x k upper triangular matrix A such that Ilall = lIaliA for every a ECk. Now

IIPnell 2 = II~ eCYizill2

t IIeaizil12 (as {ai zie } is an orthogonal sequence) i=O

(as S is an isometry) i=O

i=O

Thus IIPnellA ---+ Ilfell. The proof of (a) also shows that Pne ---+

fe = f in (H2(Ck,II.II A)· Thus,

Ilbfll M = Ilfll = Ilfell = Ilfell A = IlflI A·

• Theorem 3.3. Let A be an upper triangular k x k matrix whose diagonal elements are positive real numbers. Then the following are equivalent:

(i) A is a diagonal matrix.

(ii) There exist positive real numbers )'1, .A2, ... ,.Ak such that

for every f E H 2 (Ck).

(iii) (H2 (Ck), II.IIA) is a Hilben Ck-module.

(iv) Every closed S-invariant Ck-submodule of (H2(Ck), II.IIA) is a de Branges space.

Proof. Let the diagonal entries of A be .AI, .A2, ... , .Ak.

(i)=>(ii). As Aei = .Aiei, it is obvious that (i) =>(ii).

(ii)=>(iii). It is easy to see that (H2 (C k ),11 . IIA) is a Hilbert Ck _

module if and only if (ei, ej) A = 0 for all i f= j. Now

Ilei + ejll~ -ilei - ejll~ + i Ilei + iejll~ - i Ilei - iejll~ (.A; + .A;) - (.A; + .A;) + i (.A; + .A;) - i (.A; + .A;) o.

(iii)=> (iv). If bE Ker S* (= Me S(M)), a ECk, and gE M, then

(ab, zg) A = (b, azg) A = O.

Thus Ker S* is a Ck-submodule of H 2(Ck ). So, by Theorem 3.1, M is a de Branges space.

(iv)=>(i). Let the (i,j)th entry of A be aij. Suppose that A is not a diagonal matrix. Then there exist i,j (i < j) such that aij f= O. Choose

(iv)=?(i). Let the (i,j)th entry of A be aij. Suppose that A is not a diagonal matrix. Then there exist i, j (i < j) such that aij f. O. Choose the "first" entry aij (that is, with least j and then least i) out of the nonzero entries above the diagonal. Note that min( i, j) > 1. Then

and

Let

bz = ;/i + [~i~; ei + ;/j] z

= ßlei + (ß2ei + ß3ej) z.

Let M be the smallest closed Ck-submodule of (H2 (C k ), II.IIA) containing the set {o:znb: 0: E Ck and n 2: O}. Then S(M) c M. We are given that M is ade Branges space. Now if 0: = (0:1, ... ,O:k) and n 2: 1, then

(b, o:znb) A (ßl ei + (ß2 ei + ß3ej) z, O:ißl eiZn + (O:iß2 ei + O:jß3ej) zn+!) A

= (A (ß2ei + ß3ej) z, A (O:ißlei) zn)H2

= (A(ß2ei+ß3ej), A(O:ißlei)Zn-l)H2

This is zero if n f. 1. For n = 1, we get

(Aiß2 ei + ß3 (aijei + ai+l,jei+l + ... + ajjej) , AiO:ißl ei) H2

(Aiß2ei + ß3aij ei, AiO:ißlei)H2

\ Ai ~i~; ei + ;j aijei, AiO:i ;i ei ) H2

O.

Thus b E Me S(M). Then eib E Me S(M) by Theorem 3.1. Hence

o (ßlei + ß2eiz, ßlei z + (ß2 ei + ß3ej) Z2) A

(A (ßlei) + A (ß2ei) z, A (ßlei) z + A (ß2ei + ß3ej) Z2) H2

(A (ßlei), A (ßlei)) H2

ß2AißlAi -aij

Aj

This is a contradiction. Thus A is a diagonal matrix. •


Remark 3.4. Parts (i) and (iv) of the theorem show that there is an abundance of spaces satisfying the hypothesis of Theorem 3.1 that are not de Branges spaces. In fact, the final part of the proof gives aspace of the type b( z) H2 (Ck ) that is not a de Branges space.

Corollary 3.5. (to Theorem 3.3) Let M satisJy the hypothesis oJ Theorem 3.1. Then the Jollowing are equivalent.

(i) M is a Hilbert Ck-module.

(ii) A is a diagonal matrix.

(iii) There exists c(z) E HOO(Ck) such that {anznc(z)} is an orthogonal sequence in M Jor every sequence {an} in Ck and

Proof. It is easy to see that M is a Hilbert Ck-module if and only if (b(i),b(j)}M = 0 for all i # j. But (b(i),b(j)}M = (ei,ej)A for all i and j. Thus Theorem 3.3 gives that (i) is equivalent to (ii).

Now suppose that A is a diagonal matrix whose diagonal entries are

Al, A2, ... , Ak respectively. Then Ilb(i)IIM = Ai. Let c = I:~=1 bl:)· Then it

is obvious that c(z)H2(Ck) = b(z)H2(Ck). Now,

Ilcfll~ IIC(1)!(1) + C(2)!(2) + ... + C(k)!(k)ll~

11 :1 b(l)!(l) + :2 b(2)!(2) + ... + :k b(k)!(k) 11:

Ilb (z) (:1 J(l) + :2 J(2) + ... + :k J(k)) 11:

11 :1 J(l) + :2 J(2) + ... + :k J(k) 11:

IIJ(1) 11:2 + IIJ(2) 11:2 + ... + IIJ(k)II:2 (see Theorem 3.3(ii))

IIJIIH2.

Thus (i) and (ii) imply (iii). To see that (iii) implies (i), we note that (acJ, cg) M = (aJ, g) H2 =

(I, "iig) H2 = (cJ, "iicg) M •


Acknowledgements. The second author would like to thank the Indian Statistical Institute, Delhi Centre, for providing a stimulating mathematical atmosphere and excellent facilities while this paper was being completed. He would also like to thank the University Grants Commission for a research grant under the Career Award.

References

[1] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta. Math. 81 (1949), 239-255.

[2] F.F. Bonsall and J. Duncan, Complete Normed Algebras, Springer Verlag, 1973.

[3] L. de Branges, Square Summable Power Series, Springer Verlag (to appear).

[4] H. Helson, Lectures on Invariant Subspaces. Academic Press, 1964.

[5] K. Hoffman, Banach Spaces of Analytic Functions, Prentice Hall, 1962.

[6] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1990.

[7] M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Oxford University Press, 1985.

[8] D. Singh, Brangesian Spaces in the Polydisc, Proc. Amer. Math. Soc. 110 (1990), 971-977.

[9] D. Singh and S. Agrawal, De Branges modules in l2-valued Hardy spaces of the circle and the torus, Journal of Mathematical Sciences (U.N. Singh Memorial Volume) 28 (1994), 235-266.

[10] D. Singh and S. Agrawal, De Branges spaces contained in some Banach spaces of analytic functions, Illinois Journal of Math. 39 (1995),351-357.

[11] B.S. Yadav, D. Singh and S. Agrawal, De Branges Modules in J!2(Ck ) of the Torus IN Functional Analysis and Operator Theory, Lecture Notes in Mathematics (Number 1511), Springer Verlag (1992),55-74.


Dept. of Mathematics, St. Stephen's College, Delhi-110007, India

E-mail: [email protected]; Department of Mathematics, University of Delhi, Delhi-110007, India. Currently at Indian Statistical Institute, 7 S.J.S. Sansanwal Marg, New Delhi-110016, India

Some Methods to Find Moment Functions

on Hypergroups

Leonard Gallardo

Abstract

We present two methods to find moment functions on hypergroups. The first consists in the determination of admissible paths in the dual and the second is based on the property of asymptotic drift of the convolution. Some illustrative examples are also considered.

1. Introduction

In classical prob ability theory the usefulness of the notions of expectation and variance is due to the additivity property. Consider now a hypergroup (K, *) and suppose there exists a pair (mI, m2) ofmeasurable real-valued and locally bounded functions on K such that

(1.1) (ox * Oy, ml) = ml(x) + ml(Y)

(1.2) (ox * Oy, m2) = m2(x) + 2ml(x)ml(Y) + m2(Y)

(1.3) mi(x) :::::; m2(x),

for all x, Y E K (0 denotes Dirac measure). Such functions ml and m2 are called associated moment functions of order 1 and 2 respectively. We will say more briefly that (mI, m2) is a pair of moment functions. With respect to this pair and for any prob ability measure /-l E MI(K) on K with (/-l, m2) < +00, we define its generalized expectation and variance by

(1.4) E*(/-l) = (/-l, ml)

(1.5) V*(/-l) = (p, m2) - (p, ml)2.

For a K-valued random variable X distributed according to /-l, we also denote

14 L. Gallardo

From properties (1.1) to (1.3) it follows immediately that the operators lE* and V" have the additive property; that is, for /-Li E MI (K) such that (/-Li,m2) < +00 (i = 1,2),

(1.7) lE*(/-LI * /-L2) = lE*(/-LI) + lE*(/-L2).

(1.8) V*(/-LI * /-L2) = v"(/-LI) + V*(/-L2).

Such generalized expectations and variances were used in the 1960s by Karpelevich et al. to study a central limit theorem in the PoincareLobachevski space [Ka] and then by many other authors ([F], [Tl] etc.), but the fundamental importance of this not ion for the study of probability limit theorems on hypergroups was dearly revealed by Zeuner in 1992 [Z]. We can summarize his ideas as folIows:

Let (Sn) be a discrete time increment process on the hypergroup (K, *); that is, for each n E N and each Borel set A c K we have

lP'(Sn E AIFn-d = (8Sn _ 1 * vn)(A) a.s.,

where (Fn) is the canonical filtration of (Sn) and (Vk) is a sequence of probability measures on K. If Vk = v (independent of k), (Sn) is a random walk of law v. For such a process, if (mI, m2) is a pair of moment functions on (K, * ), the realvalued stochastic processes

and

are martingales (i.e. processes very dose to processes of sums of independent real random variables). They can then be analyzed with dassical tools in order to get information on the asymptotic behavior of the trajectories of (Sn)' For example, if (Sn) is a random walk on K of law v with v"(v) < +00, we have

lim ml(Sn) -lE*(Sn) = 0 a.s. n-..oo n

([Z], Theorem 7.2, p. 400).

The problem of how to find a pair of moment functions on a hypergroup is far from trivial. This quest ion has not yet been studied for itself. In general, moment functions are determined on each particular example

Some Methods to Find Moment Functions 15

by ad-hoc methods. In this paper we will review what can be said about this problem.

At present there are two known methods to construct moment functions on commutative hypergroups. The first one requires investigating the dual of the hypergroup. The other is more geometrical but works only on a dass of hypergroups with a convolution having asymptotic drift at infinity [G2]. For some examples the two methods give rise to the same moment functions, but the problem of uniqueness is not yet cleaL In fact, our paper does not give exhaustive answers except perhaps for the case of polynomial hypergroups where our results are probably new and the problem of the determination of moment functions is completely solved. The structure of our paper is as follows:

Section 2 contains generalities about hypergroups and some properties of moment functions, Section 3 reviews the known methods of finding moment functions and Section 4 investigates three typical examples with explicit calculations.

2. Generalities

2.1. Definitions

In this paper we consider hypergroups in the sense of Jewett [BH]. For the sake of completeness, recall that (K, *) is a hypergroup if K is a locally compact topological space with an operation *, called convolution, on the space M(K) of bounded complex Radon measures on K. The operation * is abilinear and separately continuous mapping from M(K) x M(K) into M(K) and preserves probability measures (i.e. M1(K) * M1(K) c M1(K)). Moreover, the following axioms must be satisfied:

(\Ix, y, z E K).

C2 There exists an element e E K such that be * bx = bx * be = bx •

C3 There exists a continuous involution x -+ x- of K such that e E supp(bx * by) if and only if x = y- and the canonical extension of this involution to M(K) satisfies (bx * by )- = Dy - * Dx-.

C4 The mapping (x, y) -+ bx * Dy is weakly continuous from K x K into M1(K).

C5 supp(bx * by) is compact and the mapping (x, y) -+ SUpp(Dx * Dy)

16 L. Gallardo

from K x K into the set of nonvoid compact subsets of K (endowed with the Michael topology) is continuous.

A locally compact topological group G is a hypergroup with natural convolution bx * by = bxy .

(K, *) is commutative if bx * by = by * bx(''ix, y E K). In particular, if x- = x we say that K is Hermitian; in this case it is a commutative hypergroup.

A commutative hypergroup always has a Haar measure [S], i.e. a positive Radon measure w on K such that

(2.1.1) (w, bx * 1) = (w, f),

for every f E Cc(K) and x E K where bx * f is the function defined by (bx * J)(y) = (bx- * by, f)·

For a commutative hypergroup (K, *), X(K) denotes the set of all continuous functions: x: K ---> C such that for all x, y E K

(2.1.2)

The dual k of K is then the set of those X E X(K) that are bounded and satisfy X(x-) = X(x), for all x E K. Such functions X are called characters. A Fourier analysis can then be developed, but we will not need it in this paper (see [B-H] for details).

2.2. Examples

2.2.1. Chebli-Trimeche hypergroups:

Let A be an increasing unbounded real function on lR+ such that A(O) = O. We suppose A differentiable, A'/A nonincreasing on lR~, lim A'(x)/A(x) = 2p 2: 0 and A'(x)/A(x) = a/x + B(x) in a

x ...... +oo neighborhood of zero, with a > 0, B a Coo odd function on lR. Let us consider the operator

(2.2.2) d? A'(x) d

ß = dx2 + A(x) dx·

For every Coo even function f on lR, the solution U on lR+ x lR+ of the Cauchy hyperbolic problem ßxu = ßyu, with initial conditions u(x, 0) =

f(x) and :y u(x, 0) = 0, can be written in the form

(2.2.3) r+oo u(x, y) = Jo f(t)J.Lxy(dt) ,

where J.Lxy E MI (lR+) is a probability measure with support the interval [Ix - yl, x + y] [C]. If we set Ox * Oy = J.Lxy, x- = x, e = 0, (lR+, *) with the usual topology is called the C-T hypergroup with function A. The Haar measure is w(dx) = A(x)dx and X(K) is the set of functions <PA(.\ E c) that are solutions of the eigenvalue problem

(2.2.4)

Moreover, the dual IR+ consists of all <PA E X(K) with.\ E lR+ Ui[O, p].

2.2.5. Polynomial hypergroups:

Let Pn, qn and rn be three sequences of real numbers such that Pn > O,rn ~ O,qn+1 > O,qo = ° and Pn + qn + rn = 1 for all n E N. The polynomials defined by Po == 1, PI (x) = x and

(2.2.6) (n ~ 1)

are orthogonal polynomials on [-1,1] with respect to so me measure dII(x). If their linearization coefficients are nonnegative (i.e. for all m, n,

m+n we have Pm(x)Pn(x) = L c(m, n, r)Pr(x) with c(m, n, r) ~ ° for all

r=lm-nl r), we can define an Hermitian hypergroup structure (N, *) on N with

e = ° and

(2.2.7) m+n

Dm * On = L c(m, n, r)Or. r=lm-nl

It is called a polynomial hypergroup with parameters (Pn), (qn) and (rn). 00

The Haar measure is given by w = L w(n)on with

(2.2.8) () POPI ... Pn-I wn =---

ql ... qn

n=O

(n ~ 1) (w(O) = 1).

The characters are the functions on N : n ----> Pn(x) with x E [-1,1] (see [B-H]).

2.2.9. The disk polynomial hypergroup:

Let a > 0. The disk polynomials (R'k,i(Z, Z)\k,i)EN2 are orthogonal on the unit disk D = {z E IC; Izi ~ I} with respect to the measure

18

(2.2.10) 0'+1

1fa(dxdy) = --(1 - x 2 - y2)adxdy, 1f

L. Gallardo

and are normalized by R~,i(1, 1) = 1. In polar coordinates they can

be expressed in terms of the Jacobi polynomials p/a,ß) (normalized by p/ a,ß) (1) = 1) by the formula

(2.2.11)

By a result of Koornwinder [Ko] , these polynomials satisfy a linearization formula with nonnegative coefficients:

(2.2.12) R~l,il R~2,i2 = L ca(kl , f l , k2 , f 2 ; k, f)R~,i' (k,i)

where the summation is over pairs (k, f) such that

(2.2.13) kl + k2 + f = f l + f 2 + k and Ikl + f l - k2 - f 21 ~ k + f ~ kl + f l + k2 + f 2 .

We can then define a convolution on N2 by

(2.2.14) O(k1,il) * O(k2,i2) = L ca(k l , f l , k2 , f 2; k, f)O(k,i). (k,i)

We then get a commutative hypergroup (N2 , *) with e = (0,0) and involution (k, f)- = (f, k). It is called the disk polynomial hypergroup [B-G] or the dual disk hypergroup ([B-H] or [Z]).

2.3. General properties of moment functions

Let (K, *) be a hypergroup. We will denote by MI the set of all real valued measurable and locally bounded functions ml on K satisfying (1.1), i.e. MI is the set of all moment functions of order 1. We denote also by M the set of all pairs of moment functions on K. We begin this section with some information about the size of M. The first result is useful even if its proof is trivial.

2.3.1. Proposition. (a) MI is areal vector space (for the usual addition of functions and scalar multiplication).

(b) If (mI, m2) and (mI, m2) belong to M then m2 - m2 E MI. Conversely, if (mI, m2) E M then (mI, m2 + md E M for every ml E MI such that mi ~ m2 + ml.

We now consider the example of a topological group.

2.3.2. Proposition. If C is a locally compact topological group, then every pair of moment functions on C is of the form (mI, mr) for some ml E MI.

Proof. Let (ml,m2) E M. For every x E C, we have ml(xx-1) = ml(x) +ml(x-l) = ml(e) = 0 so that ml(x-l ) = -ml(x). This implies, by using (1.2) with y = x-I, the following equation:

m2(x) + m2(x-l) = 2(ml(x))2,

using again m( e) = O. By eondition (1.3) we deduee immediately that m2(x) =m2(x-l ) = (ml(x))2. •

2.3.3. Remark. If C = ]Rd with its additive strueture and usual topology, (2.3.2) applies but more ean be said. In this ease MI eoineides with the spaee of eontinuous linear functionals on ]Rd. Indeed, letting ml E MI, we have ml( -x) = -mI (x) and ml(x+y) = ml(x)+ml(Y) for all x and y in ]Rd. But this implies eontinuity of ml by classieal arguments (reeall that we have assumed measurability and loeal boundedness of ml)' Finally, eontinuity of ml implies ml (o:x) = o:ml (x) for every x E ]Rd

and 0: E ]R.

The property of having moment functions of the form (mI, mi) is in a eertain sense a eharacteristie of groups.

2.3.4. Proposition. Let (K, *) be a hypergroup. Suppose there exists a pair of moment functions of the form (mI, mi) with ml E MI and ml(x) =I 0 for all x =I e. Then (K, *) is a topological group.

Proof. We have (Dx * Dy , mr) = (Dx * Dy , ml)2, which implies that ml is a eonstant on supp(8x * 8y) for every x, y E K. In particular ml(u) =

ml (e) = 0 for every u E supp( 8x * 8x -) beeause e E supp( 8x * 8x - ). This implies supp( 8x * 8x-) = {e}, i.e.:

(2.3.5) (Vx E K).

Let now x, Y E K and fL = 8x * 8y. We have to prove that fL is a Dirae measure. By assoeiativity of the eonvolution we have 8x - * fL = 8y , so that for every bounded measurable function f on K we have

(2.3.6) (fL,8x * f) = f(y)·

Choose f such that f(y) = 1 and 0 :S f(u) < 1 for all u =I y. From (2.3.6) it follows that the funetion bx * f is equal to 1 fL almost everywhere. But 8x * f(t) = 1 means (bx - * bt , f) = 1 so f = 1 bx - * bt a.e. and from the special form of f we deduee that bx- * 8t = by . Multiplying both sides of this equation by bx gives 8t = bx * by • •

20 L. Gallardo

2.3.7. Remark. If the condition ml(x) i= 0 for every x i= e is not assumed, the result of the proposition fails beeause ml ean be identieally o on a subhypergroup of (K, *). Take for example K = G x C, the direet produet of a topologie al group by a eompact hypergroup, and (mI, mD a pair of moment functions on G. Then if we set

ml(X,y) = ml(x)

for every (x, y) E G x C, it is easily verified that (mI, mD is a pair of moment functions on (K, * ), and (K, *) is not a group. We now eome to the eontinuity properties of moment funetions.

2.3.8. Proposition. ([Z] 5.12 and 5.21): Let (ml,m2) be a pair oj moment junctions on a hypergroup (K, *) with a Haar measure w. Then ml and m2 are continuous junctions on K.

Proof. Let 9 be a eontinuous and eompaetly supported function on K. By axioms C4, C5 and by Lebesgue's dominated eonvergenee theorem, it is easy to see that the function

X f---t fK ml(Y)(ox- * g)(y)w(dy)

is eontinuous on K. But it is also equal to

and the eontinuity of ml follows. A similar argument using (1.2) gives the eontinuity of m2. •

2.3.9. Remarks

(i) A non-identically-zero first-order moment function ml eannot be bounded on K ([Z], 5.13). Indeed suppose sup ml = a < +00 and

K

inf ml = b > -00. For every E > 0, choose XE E K such that ml(xE ) > K

a - E. Then we have

a ~ (ox< * Oy, ml) = ml(xE ) + ml(y) > a - E + ml(y)

for every y E K. Hence ml (y) ::; E which implies ml ::; o. A similar argument using b gives ml ~ o. Then ml == o.

(ii) In the same manner we ean show that, if (mI, m2) E M and m2 is bounded on K, then m2 == 0 and ml == O. The loeal boundedness of

ml and m2 implies that, on a compact hypergroup, moment functions are identically zero.

(iii) Another example of a hypergroup on which moment functions are always trivial is furnished by a connected simple (nontrivial) topological group G. It is an easy exercise to prove in this case that if ml E MI then ml := 0.

3. Determination of moment functions

In this section we suppose that (K, *) is a commutative hypergroup.

3.1. Admissible paths in the structure space

Let t t---+ <Pt be a map from [0,1] into X(K) such that <Po := 1 and the derivatives

(3.1.1)

and

(3.1.2)

exist locally uniformly in x and are real-valued. We will then say that t t---+ <Pt is an admissible path in X(K). By the Leibniz differentiation rule it is easy to see that ml and m2 are measurable locally bounded functions on K that satisfy conditions (1.1) and (1.2).

Condition (1.3) is unfortunately not always true but there are some interesting situations in which (1.3) holds. In order to introduce them, note that typical examples of real-valued functions f on [0,1] satisfying

(3.1.3) (1'(0))2:::; 1"(0)

are f(t) eßt (ß E R) and convex combinations of such functions, i.e. f(t) = 2::~1 o:;eßit with 0:; ~ O. More generally, Laplace transforms f(t) = IR eßtl/(dß) , where 1/ is for example a compactly supported positive measure, also satisfy (3.1.3) by Cauchy-Schwarz inequality. These trivial remarks apply to the cases where admissible paths in X(K) have a Laplace representation of the form

(3.1.4)

where for each x E K, I/x is a compactly supported probability measure on R. In this case (mI, m2) E M.

22 L. Gallardo

3.1.5. A trivial example in which a representation (3.1.4) holds is the case of ]Rd. Let ml E MI (see 2.3.3); 'Pt(x) = etm1 (x) is an admissible path for which Vx = omdx).

3.1.6. For C-T hypergroups (see 2.2.1) we know [T2] that there exists a continuous kernel K (x, u) such that

(3.1. 7) 'P-\(x) = LXx K(x, u) cos(Au)du (A E IC).

We immediately see that

t 1----+ 'Pi(t+p) (x) = L: K(x, u) cosh((t + p)u)du

is an admissible path with a Laplace representation. The associated pair (mI, m2) of moment functions was given in ([Z] (5.17)):

(3.1.8)

(3.1.9)

{X (Y A(z) ml(x) = 2p Ja Ja A(y) dzdy

2 {X (Y rz r A(z)A(v) {X (Y A(z) m2(x) = 8p Ja Ja Ja Ja A(y)A(u) dvdudzdy + 2 Ja Ja A(y) dzdy.

3.1.4. Remark. The "path method" in X(K) to obtain moment functions requires a Laplace representation of the multiplicative functions. In the theory of special functions this is considered a difficult problem. In general, this method is hard to handle especially in the case of polynomial hypergroups (see the examples studied in [V]).

3.2. The case of asymptotic drift of the convolution (ADe) Let (K, *) be a commutative hypergroup with K an unbounded sub

set of ]Rd carrying the usual topology (the discrete topology if K is a lattice). We will suppose that the convolution is a moderate perturbation of the additive group structure; that is, it satisfies the following conditions [G2]:

(H) For some norm 1.1 on ]Rd, some constant C > 0 and for every x, y E K and u E supp(ox * oy), we have

lu - yl ::; Clxl·


Now let L be a eontinuous linear form on Jl~d and eonsider the funetions on K x K defined by

(3.2.1)

(3.2.2)

mt(t, x) = (Dt * Dx , L - L(t))

m2(t,x) = (Dt * Dx , (L - L(t))2).

3.2.3. Definition. We say that the eonvolution * has an asymptotie drift relative to the linear form L if the limits

(3.2.4) lim mt(t, x) = mt(x) t~oo

(3.2.5) lim m2(t,x) = m2(x) t~oo

exist for all x E K (t ----t 00 me ans Itl ----t +00 and t E K).

We will abbreviate this by saying that (K, *) is an ADC hypergroup without referenee to the linear form L when there is no risk of eonfusion. (This terminology is inspired by the theory of stoehastie proeesses.) For example, eondition (3.2.4) indieates that the drift in the direetion L of the random walk on K with law JL = Dx tends to a limit at infinity; see [G1] or [G2] for details.

3.2.6. Theorem. 1f (K, *) is an ADe hypergroup, formulas (3.2.4) and (3.2.5) define a pair (mt, m2) of moment functions on K.

The proof of this result is based on measure theoretie arguments (see [G2] for all the details). Let us only mention that property (1.3) here is trivial to obtain. Indeed by the Cauehy-Sehwarz inequality we immediately get mi(t, x) ::; m2(t, x) for all t and x E K, and mi(x) ::; m2(X) follows by letting t ----t 00.

3.2.7. Remark. The result of the theorem remains true if we suppose that t tends to infinity with some direetional restrictions instead of isotropie eonvergenee (see [G2] for details).

3.2.8. Examples

(i) In the ease of JRd, the quantities mt(t,x) and m2(t,x) are independent of t and respeetively equal to L(x) and (L(X))2.

(ii) For the Tehebyehev polynomial hypergroup (i.e. Pn ( eos B) = eos nB) the eonvolution is given by Dt * Dx = HDlt-xl + Dt+x ) for t, x E N. In this ease mt(t,x) = ~(It - xl + x - t) (= 0 if t > x) and m2(t,x) = ~((It - xl - t)2 + (t + x - t)2) (= x2 if t > x) and we have an ADC hypergroup (with L( u) = u). The eorresponding moment functions are mt(x) = 0 and m2(x) = x2.

24 L. Gallardo

(iii) The Bessel Kingman hypergroup is a C-T hypergroup with function A(x) = x2"+1(a > -1/2), and its convolution can be written in the form

(3.2.9) (8t * 8x f) = [11 f((t2 + x2 + 2txZ)1/2)dF,,(z),

r(a+1) 21/2 where dF,,(z) = J7fr(a + 1/2) (1 - z )"- dz is a probability measure

[Ka]. For L( u) = u, we have

(3.2.10) mi(t, x) = [11 ((e + x2 + 2txZ)1/2 - t)idF,,(z) (i = 1,2).

It is easy to see that the limits lim mi(t, x) exist and are respectively t ..... +oo

m1(x) = 0 and m2(x) = c"x2 (c" > 0 is a constant).

3.2.11. Generalized expectation and variance

Suppose K C Rand (K, *) is an ADC hypergroup with moment functions (m1, m2) given by (3.2.4) and (3.2.5). Let us consider the associated generalized expectation E* and variance v: as defined in (1.4) and (1.5) and denote by E and Var the classical expectation and variance on R. The following result gives a qualitative relation between these classical and generalized not ions (see [G2] for details).

3.2.12. Proposition. Let J.l E M1(K) be a probability measure. (a) IfE(J.l) exists then E(8t * J.l) and E*(J.l) exist (Vt E K) and

(b) IfVarJ.l < +00 then Var(8t * J.l) < +00 (Vt E K), V*(J.l) exists and

4. Explicit moment functions in the classical cases

4.1. C-T hypergroups

We determined a pair of moment functions in Section 3 by the method of admissible paths in X(K). The method of ADC hypergroups also works in this case. We quote the following result from [G2]:


4.1.1. Proposition. C-T hypergroups are ADC (with respect to L(u) = u) and the associated moment functions m1 and m2 in the sense of (3.2.4) and (3.2.5) are the same as those given by (3.1.8) and (3.1.9).

The idea of the proof is the following: Let 9t(X) = (8t * 8x , L). It can be proved that 9t(U) = (8t * A'/A)(u) because ~ commutes with generalized translations. N ow limt~oo (8t * A' / A) (u) = 2p and we can use Lebesgue's convergence theorem as t -+ 00 in the integral

9t(X) - t = 10x (iX A~:)) (~9t)(u)A(u)du. A similar but more sophisticated argument holds to obtain m2 (see

[G2] for furt her details).

4.1.2. Remark. The question "Why do the two methods give the same moment functions?" is open. As noted in Proposition 2.3.1 M 1 is a vector space; we would like to prove that dimM 1 = 1. Let m1 E M 1. If we could prove m1 is COO ,m1(0) = m~(O) = 0, then u(x,y) = (8x * 8y ,m1) would satisfy ~xu = ~yU for all x, y. This would imply ~m1 = C (constant) and the only solution of this equation (with m1(0) = m~(O) = 0) is m1(x) = C ft A(U)-1 fou A(Od(~)du. We can prove that m1 is Coo by the following argument suggested by K. Trimeche: Let v be a Coo even function on JE. with compact support such that (m, v) = 1. Then m1 * v is of class Coo and we have

ml *v(x) = 1000 (8x *ml)(y)v(y)A(y)dy = ml(x) + 1000 v(y)ml(y)A(y)dy.

This clearly proves the result.

4.2. Poly no mi al hypergroups

Let (N, *) be a polynomial hypergroup as in Section 2.2.5. Let us m

denote Pm(x) = L aimlxk . From the definition of the convolution we k=O

deduce the crucial representation

(4.2.1) (\fm E N),

where in the expression of Pm, x k is replaced by 8t = 81 * ... * 81 (k times). Let us begin with an easy fact concerning the derivative of the polynomials Pn at the point 1.

4.2.2. Proposition. positive numbers.

(P~(l))n>O is a strictly increasin9 sequence of

26 L. Gallardo

Proof. From (2.2.6) we immediately deduce that the sequence U n = P~ (1) satisfies

(4.2.3) (n ~ 1),

with Uo = 0 and Ul = 1. We can rewrite (4.2.3) in the form

(4.2.4)

and the result follows clearly by induction. • 4.2.5. Proposition. If a polynomial hypergroup has a pair (ml, m2) of moment functions, then there exist a E JR and b ~ 0 such that

{4.2.6}

{4·2.7}

ml(n) = aP~(1)

m2(n) = a2 P:(I) + bP~(1). Conversely, if there exist constants a E JR and b ~ 0 such that

{4·2.8}

jor all nE N, then the hypergroup has moment junctions given by {4.2.6} and (4.2.7).

Proof. Let (ml, m2) be a pair of moment functions. By (1.1) and iteration, for every k E N* and yEN, we have

n

We deduce that, for every polynomial P(x) = L (XkX\

k=l

n

L (Xk(8~ * mt}(y) k=l

P'(l)ml(1) + P(l)ml(Y)·

With P = Pn and using (4.2.1) we immediately obtain (4.2.6) with a = ml(l). To get (4.2.7) consider the associated variance V* given by (1.5). For every k E N, we have

(4.2.9)

Multiplying this relation by (Xk and summing over k = 1,·· ., n, we obtain

(P(od * m2)(O) - (P"(1) + P'(1))mi(1) = P'(1)V,(od, n

where P(x) = L CXkxk. If P = Pn, using (1.2) and (4.2.1), and taking k=1

into account that V,(od = m2(1)-mi(l) and ml(O) = 0, we immediately get (4.2.7) (with a = ml(l) and b = m2(1)).

For the converse, let ml and m2 be defined by (4.2.6) and (4.2.7). Assumption (4.2.8) is condition (1.3). Let us verify (1.1). For every yEN we have

Pyml(Y + 1) + ryml(Y) + qyml(Y - 1) a(P~(1) + 1) = ml(Y) + a.

By induction we obtain (of * ml) (y) = ml (y) + ka, and this implies that

(on *md(y) = (Pn(OI) *ml)(Y) = ml(Y) + (E CX~k) a = ml(Y) +ml(k).

The same arguments also work to verify condition 1.2. •

4.2.9. Remark. Condition (4.2.8) is always satisfied with a = 0 and b > 0 but in this case ml == O. The pair (mI, m2) with ml == 0 will be called a trivial pair of moment functions. The following result is a direct consequence of 4.2.5.

4.2.10. Corollary. A polynomial hypergroup has a non-trivial pair of moment functions if and only if

P~(1)2 - P:;(I) sUPn>O P~(I) < +00.

In this case with the notations of (2.2.1) we have dimM I = 1 and the uniqueness of m2 modulo MI is also guaranteed.

4.2.11 Example

The Tchebychev polynomial hypergroup (i.e. with Pn ( cos B) = cosnB) has only trivial moment functions (i.e. ml == 0 and m2(n) = bP~(I)). Indeed, we have P~(1) = n2 and P:;(I) = 1/3(n4 - n2), and the sequence P~(1tl(P~(1)2 - P:;(1)) tends to infinity.

4.2.12. Remark. Explicit expressions in terms of the parameters Pn, qn, rn for the sequences P~(1) and P:;(I) are not difficult to obtain.

28 L. Gallardo

If we differentiate (2.2.6) we see that the sequences U n = P~(1) and Vn = P:(l) satisfy the recurrence relations (4.2.3) and

(4.2.13) (Vo = VI = 0).

If we let D.k = Uk - Uk-l, we get from (4.2.3) the linear difference equation

(4.2.14)

where ak = qk/pk and bk = l/pk' By finite induction we obtain

(4.2.15)

and this immediately determines Un by summing D.k over k = 1,··· , n. In the same manner D.k = Vk - Vk-l satisfies an equation like (4.2.14) with ak = qk/pk and bk = 2Uk/pk, and the result is analogous to (4.2.15); we omit the details.

We will now give an example of a dass of polynomial hypergroups with asymptotic drift of the convolution. We will say that a polynomial hypergroup has converging parameters ifthe limits limn~CXl Pn = P E]O, I[ and limn~CXl qn = q E]O, I[ exist.

4.2.16. Proposition. A polynomial hypergroup with converging parameters is an ADe hypergroup (with L(u) = u), and the moment junctions according to (3.2.4) and (3.2.5) are given by

(4·2.17)

(4·2.18)

ml(n) = (p - q)P~(I)

m2(n) = (p + q)P~(l) + (p - q)2 P:(l).

Proof. (see [G2] for all details): We will give only the main ideas. We have (8x * 81, L - x) = Px - qx -t p - q as x -t +00. By induction, for every k E N, we obtain

lim (8x * 8~, L - x) = k(p - q). x----++oo

Then using (4.2.1) we deduce

In the same way we can prove that

lim (ox * or, (L - X)2) = k(p + q) + k(k - l)(p _ q)2 x->co

and using (4.2.1) again we get the formula for m2(n). • Remark. A polynomial hypergroup with converging parameters such that p > q has nontrivial moment functions. Using Corollary 4.2.10, this shows indirectly that the condition sUPn>O(P~(1))-1(P~(1)2 - P::(l)) < +00 is satisfied. We don't know if this is easy to verify by analytical methods.

4.2.20. Remark. Prom formula (4.2.15) and the expression for the Haar measure given in (2.2.8) we can rewrite P~(1) in the form

(4.2.21) P~(l) = t ~ w(j) = t w([O, k -1]). k=lj=Ow(k)qk k=l W(k)qk

This establishes a link between ml (n) and the Haar measure analogous to the case of C-T hypergroups where we have found ml(x) =

2p lax A~y)w([O,Y])dY (see (3.1.8)). This remark is a very incomplete

answer to a question of Prof. Ajit Iqbal Singh concerning the relations between moment functions and Haar measure. This remains an interesting and open problem.

4.3. Moment functions on the disk polynomial hypergroup

From the structure of the support of O(m,n) * 8(k,l) (see (2.2.13) and (2.2.14)) we can see easily that the linear form L(u, v) = u-v is constant on supp(o(m,n)*O(k,l))' This shows that the functions ml(t, x) and m2(t, x) do not depend on t and so the convolution has asymptotic drift relative to L with moment functions

(4.3.1)

We quote without proof the following result from [G2], which gives another pair of moment functions.

4.3.2. Proposition. Convolution on (N2, *) has asymptotic drift relative to the linear form L(u, v) = u+v when t -+ 00 in the direction ofthe half line u = v (see Remark 3.2.7). The corresponding moment functions are given by

(4.3.3)

30 L. Gallardo

4.3.4. Remarks

(i) Moment functions (4.3.3) were used in [B-G] to obtain a law of large numbers on (N2 , * ).

(ii) The problem of determining all possible pairs of moment functions on the polynomial disk hypergroupis open, but in our opinion a generalization of the methods used in Section 4.2 to multidimensional polynomial hypergroups is possible.

References

[B-G] BOUHAIK, M., GALLARDO, L. [1992] Un theoreme limite central dans un hypergroupe bidimensionnel. Ann. lnst. Henri Poincare Prob. Stat. 28 n° 1, p. 47~61.

[B-H] BLOOM, W., HEYER, H. [1995] Harmonie analysis of prob ability measures on hypergroups. Walter de Gruyter Ed., Berlin-New York.

[C] CHEBLI, H. [1974] Operateurs de translation generalisee et semi groupes de convolution. Lecture Notes in Math. 404, p. 35~59.

[F] FA RAUT , J. [1975] Dispersion d'une mesure de probabilite sur 8L(2, lR) biinvariante par 80(1, lR) et theoreme de la limite centrale. Universite Louis Pasteur - Strasbourg.

[GI] GALLARDO, L. [1996] Chaines de Markova derive stable et lois des grands nombres sur les hypergroupes. Ann. Inst. H. Poincare, Prob. Stat., a paraitre.

[G2] GALLARDO, L. [1996] Asymptotic drift of the convolution and moment functions on hypergroups. Math. Zeitschrift, to appear.

[Ka] KARPELEVICH, F.L, TUTUBALIN, V.N., SHUR, M.G. [1959] Limit theorems for the composition of distributions in the Lobachevsky plane and space. Theory of Prob. and Appl. 4, p. 399~402.

[Ko] KOORWINDER, T.H. [1978] Positivity proofs for linearization and connection coefficients of orthogonal polynomials satisfying an addition formula. J. London Math. Soc. 28, p. 101~114.

[Ki] KINGMAN J.F.C. [1963] Random walks with spherical symmetry. Acta Math. 109, p. 11 ~53.

[S] SPECTOR, R. [1978] Mesures invariantes sur les hypergroupes. Trans. Amer. Soc. 239, p. 147~165.

[Tl] TRIMECHE, K. [1978] Probabilites indefiniment divisibles et


theoreme de la limite centrale pour une convolution generalisee sur la demi droite. Comptes Rendus Acad. Sc. Paris, Serie A, t. 286, p. 63-66.

[T2] TRlMECHE, K. [1981] Transformation integrale de Weyl et theoreme de Paley-Wiener associes a un operateur differentiel singulier sur [0,00[. J. Math. Pures-Appl. 60, p. 51-98.

[V] VOlT [1990] Laws of large numbers for polynomial hypergroups and some applications. J. Theor. Prob. 3, p. 245-266.

[Z] ZEUNER, H. [1992] Moment functions and laws of large numbers on hypergroups. Math. Zeitschrift 211, p. 369-407.

Universite de Brest, Departement de Mathematiques, 6 Avenue Le Gorgeu BP 809 29285 BREST - FRANCE.

From September 1st 1996: Universite de Tours, Departement de Mathematiques, Faculte des Sciences et Techniques, Parc de Grandmont 37200 TOURS - FRANCE

About Some Random Fourier Series and Multipliers Theorems on

Compact Commutative Hypergroups

Marc-Olivier Gebuhrer

Abstract

The character theory of compact commutative hypergroups is yet far from being weIl understood; as this paper shows, the behaviour of the Plancherel measure is related to some deeper harmonie analysis involving notably Sidon sets. Pioneering wark related to this area has been performed a long time aga by R. Vrem, K. Ross, J. Fournier (see references below). This elementary study provides only a sampie of easy results, but paves the way to apparently much deeper questions.

This paper is comprised of two parts: the first one has been already mentioned; the second provides some easy multiplier theorems. In the context of hypergroups, and despite its elementary character, much of the material presented here is new.

1. General notations; preliminary results

Let X be a compact topological Hausdorff space; we denote by M(X) the Banach space of complex valued Radon measures on X, endowed with its natural dual norm 1111111 = sup{I(Il, J)I; 1111100 ::; 1,1 E C(X)} where C(X) stands far the Banach space of complex valued continuous functions on X.

The following definition recalls our not ion of compact commutative hypergroup as given in [Gh-

Definition 1.1. Let us assurne that (M(X), 11 . 111) is endowed with a commutative Banach algebra structure. We denote the product on M(X) by * and call it thereafter convolution on M(X).

Assurne furt her that an involutive homeomorphism x I-t X is given on X. We shall say that (M(X), *, V) is a commutative hypergroup on the compact space X provided the following requirements are fulfilled:

(NA l ) The mapping (x,y) I-t 8x * 8y from X x X into M(X) is w* -continuous.

(N A2 ) The convex set Ml(X) of probability measures on X is a semigroup for the convolution.

34 M. O. Gebuhrer

(UI ) There exists a unique point e E X sueh that 8e is the unit of M(X) and moreover e E Supp(8x * 8y ) {:} x = y.

(U2) For eaeh x, y E X such that x =1= y, there exist neighborhoods W(x), W(y) of x, y respectively sueh that e ~ clos(W(x) * W(Y)) (where clos denotes the closure in X of the braeketed set, and where, as usual, for two subsets A, B of X, we denote A * B:= UaEA Supp(8a * 8b).

bEB

1.2. We shall denote by er the unique translation invariant probability measure on X. Let us reeaIl that for f E C(X), we let

Txf(y) = (85; * 8y,!) and that (T, Tx!) = (T,!) for x E X, fE C(X).

It is weIl known that Supp( T) = X. Let us also recall that LI(X, er) is a closed ideal in M(X) and that

the following property holds:

(f.er) * (g.er) = h.er

where

h(x):= i f(Y)Tyg(x)der(y) = i g(Y)Tyf(x)der(y) for f, gE LI(X, er)

the former equalities being understood almost everywhere w.r.t. er. We denote by X the speetrum of the eommutative Banaeh algebra LI(X, er), whieh is known to be symmetrie, X being eompaet [8].

It is also classical that for the natural topology on the Gel'fand speetrum of the Banaeh algebra LI(X, er), X is a diserete topologieal space; [B-H], [8]. The eomplex vector spaee generated by elements of X, viewed as eontinuous functions on X, will be ealled the spaee of trigonometrie polynomials, and denoted by P(X). We define the Fourier transform of f E LI (X, er) by the formula:

!(X):= Ix f(x)X(x)der(x).

The Fourier transform extends to measures on X in the obvious way. We shall use repeatedly the following fundamental results:

Theorem 1.3. (Plancherel). The Fourier transform f 1-+ ! restrieted to L2(X, er) yields an isometrie isomorphism from L2(X, er) onto f2(X, w) where w is the Planeherel measure on X defined by

w{X} = Ilxll2"2 for XE X.

Random Fourier Series and Multipliers Theorems 35

Theorem 1.4. (Fejer approximation). There exists a bounded approximate unit {SOC>}C>EA in L1(X,a) such that

(i) for each a E A, SOC> E P(X), (ii) for any c > 0, there exists a cofinal subset B(c) cA such that

IlsoC> Ih ~ 1 + c for any a E B(c).

The proof of these two results may be found in [B-H] for 1.3 and [Gb for 1.4. The reader might also be interested in the forthcoming paper [G - Beb where the whole theory will be detailed within our axiomatics.

Theorem 1.4 has also been proved previously in a completely different way by[V]l. The methods cannot be compared; the approximate identities thus constructed are different. [Gb provides an effective and universal way to compute such an approximate identity on a compact hypergroup, and the methods allow generalizations to the locally compact case. [V]l is limited to the compact case and cannot be further extended.

1.5. Standing assumption for the rest of the paper. From now on we assume the space X to be metrisable.

1.6. Abuse of language. We shall speak of the hypergroup X instead of (M(X), *, V).

2. Changing signs of Fourier coefficients

We begin by mentioning the following

Theorem 2.1. (Hausdorff-Young). Let p E [1,2], p' such that 1+1..=1. P p'

(i) ff fE LP(X, a) then f E CP' (X, w) and Ilfllp ' ~ Ilfllp •

(ii) ffcjy E CP(X,w), there exists <1> E LP'(X,a) such that

M oreover the senes

L cjy(Xn)xnw{Xn} converges in U' (X, a) to <1>. n

36 M. O. Gebuhrer

Proof. It follows the classical lines and has been proved previously in general by [vh. Just notice that by defining the linear mapping T by

Tf = f for f E Ll(X,a)

and working with Ll(X, a) and coo(.X, ro), it is plain that T is of type (1, (0). By 1.3, T is of type (2,2). The result follows by applying the Riesz-Thorin interpolation theorem. •

Remark 2.1.1. By 2.1, for pE [1,2], the linear mapping f 1--+ fis continuous from

LP into Cp'; in other words !!f!!p' :::; Apllfll p for some positive Ap :::; 1. We do not know the best possible estimate for Apo We do not know how Ap depends on the hypergroup. We do not know which functions in LP are extremal with respect to that property.

Theorem 2.2. Let cjJ a complex valued function on X and let p E [1,00].

Let us call Cp(cjJ) the following property:

n

Then (i) Cl(cjJ) holds if and only if cjJ E Cl(X, ro). (ii) 1f pE [1,2] and if Cp(cjJ) holds, then cjJ E .e2 (X, ro). (iii) 1f pE [2,00] and if cjJ E .e2 (X, ro) then Cp(cjJ) holds. (iv) 1f p E]l, 2[ and if cjJ E CP(X, ro) then Cp(cjJ) holds.

Proof. In what follows property Cp(cjJ) will be abbreviated to Cp. (i) Let fl be a complex valued function on X such that

limxn-->oo !fl(Xn)! = O. We denote by co(X) the space of such functions and endow it with

its natural Banach space norm IIfllloo = sUPn W(Xn)!. The linear functional Tt/Jg defined on Ll(X, a) by the formula

n

is bounded on Ll(X, a) for any fl E Co(X); in fact, let (Tfg, f) :=

L:n<N cjJ(Xn)fl(Xn)f(Xn)ro{Xn}' We see at on ce that the linear function~ls Tfg are bounded on Ll(X, a) and assumption (Cl) teIls us

that limN-+=(T~, J) exists for any f E Ll(X, a). Our claim follows from the Banach Steinhaus theorem. But in fact, (Cl) entails that SUPllglI",,91(Tq,g,J) I< +00 for any f E Ll(X,a), as one immediately checks; by applying the uniform boundedness principle, we see that sup 1I§1I",,9 IITq,§11 < +00 where, as usual, IITII stands for the norm of

§ECo(X)

the bounded linear functional T on L 1 (X, a). Let N be a fixed integer; we choose gE Co(X), 11911= :S 1 such that

for n ::; N (as usual if 1J(Xn) = 0, we let g(Xn) = 0); now, in statement 1.4, it is always possible to replace the given bounded approximate identity by a bounded approximate identity (Kk) with the same properties, satisfying moreover the condition k k 2:: ° on X; for this, its sufficient to let K k = 'Pk * 'Pk, where for any f we define j(x) = f(x)(x EX).

This choice being done, one has

where the right written term is I (Tq,§, K k) I. Therefore

And finally as IIKklh ::; 1 + l/k, one has

By letting k -+ 00, we get

L 11J(Xn)lw{Xn} ::; sup{IITq,§II; 11911= ::; 1, gE Co(X)}. n'5:.N

The right hand term is now independent of N and this shows that 1J E fl(X, w).

(ii) P E [1,2] and (Cp ) holds. Cases p = 1,2 are trivial; case p = 1 results from (i) and the fact that fl(X, w) C f2(Xl, w). Case p = 2

38 M. O. Gebuhrer

results from the fact that L2(X, a) is a Hilbert space and Theorem 1.3 (Plancherel). Let p E]l, 2[. By using the same device as in (i), one sees immediately that the linear functional (Tq" fl = L:n </>(Xn)!(Xn)W{Xn} is bounded on LP(X, a).

Therefore,

I (Tq" fll :::; Cpllfll p for some positive Cp, fE LP(X, a).

But L2(X, a) c LP(X, a) for p E [1,2] and 11 flip :::; IIfl12 for f E L2(X, a). Therefore, Tq, defines a bounded functional on L2(X, a). There exists a unique E L 2 (X, a) such that Ix f(x)(x)da(x) =

(Tq" fl for f E L 2 (X, a). It is plain that = </> and by 1.3 again </> E p2(X, w).

(iii) If p E [2,00] and if </> E p2(X, w), then (Cp ) holds: in fact, in that case LP(X, a) c L2(X, a); so by 1.3 again the series L:n </>(Xn)!(Xn)W{Xn} is absolutely convergent for any f E LP(X, a) and this is the content of (Cp ).

(iv) If P E]1,2[ and </> E PP(X, w) then (Cp ) is true. In fact by Theorem 2.1, there exists a E LP' (X, a) (~ + ? = 1) such that

 = </>; so consider the linear functional f f--t Ix (x)f(x)da(x); we

know that the series L:n </>(Xn)XnW{Xn} converges in LP' (X, a) again by 2.1, so that

the right term being convergent for every f E LP(X, a). Now, again by 2.1, ! E pp' (X, w), so that the right hand series is

in fact absolutely convergent for any f E LP(X, a). This is the content of (Cp ). •

2.3. Given the preceding result, the following quest ion is in order: Let p E]2, 00]: does property Cp (</» imply </> E p2(X, w)? The

answer is known to be positive for any commutative compact group C, but the proof is much deeper [H]. The answer for compact commutative hypergroups is yet unknown, but important steps towards the answer will be found in the forthcoming paper [G]. Instead of turning towards that problem, we move on to explore another, intimately connected area of problems.

Definition 2.4. The Rademacher functions (rm)mEN are the functions defined on the interval [0,1] by the formulae rm(t) =

sgn sin 2n +17rt(m E N).

Classical properties of the sequence (rm ) are to be found e.g in [Z]. They will be used here without proof.

2.5. Let cP E .e2 (X, ro); for t E [0,1] we let

n

The meaning of the right-hand side series is not completely trivial. First of all, for a.e. t E [0,1], ft(x) E L2 (X, Ci) and the series

converges in L2 (X, Ci) to ft; moreover, at least farmally, for the moment,

it is not immediately clear that for fixed x EX, t f-+ ft (x) E L 2 ([0, 1], dt) and this is definitely not true in general for x = e.

However, let r(x) := Lm ICP(XmWIXm(xWro2 {Xm} for x EX, the sum being finite or not; we have

because ro{Xm} = IIXml122 so that r(x) < +00 for almost every x E X with respect to Ci. Moreover

r1 Ift(x)1 2 dt = L ICP(XmWIXm(x)1 2 ro 2 {Xm} far almost every x E X, Jo m

with respect to Ci. Much more is true in fact.

Theorem 2.6. Let cP E .e2 (X, ro). For almost every t E [0,1], the series

n

converges almost everywhere on X with respect to Ci.

Proof. Let r = {(x, t) E X x [0,1]; ft(x) is convergent}; r is Ci ® m measurable in Xx [0,1] where m denotes Lebesgue measure on [0, 1]. By 2.5 there exists a subset Eo c X, with Ci(Eo) = 0, such that for every

40 M.O. Gebuhrer

Xo E X\Eo, m{ t E [0,1]; rn {(xo, tn i- 0} = 1. In fact, we showed in 2.5 that for such an xo, the series Ln 1</J(Xn)12IXn(xo)12w2{Xn} < +00 and the assertion is just a classical result for Rademacher functions ([Z], 5.5 (i)). So, the measure of r relative to CT 0 m is 1; therefore, by Fubini's theorem, CT{r n {(x, ton i- 0} = 1 for almost every to and this is our claim. •

2.6.1. Remark. Theorem 2.6 is just an extension of the classical case as found in [Z,

5.6 (i)]. However the result is relatively unexpected given the various possible growths of the Plancherel measure w.

2.7. At this point, we make the following elementary observation: starting with </J E f2(X, w), the series Ln 1</J(Xn)l2IXn(x)l2w{Xn} converges for every x E X and even,

n

n

while, at the same time,

L 1</J(Xn)1 2 IXn(xWw2 {Xn} < +00 for almost every x E X. n

The exceptional set in the last written series is of course dependent on </J. Two natural cases present themselves (we are going to give some results for the easiest one) namely

Case (A) SUPn w{Xn} < +00

Case (B) sUPn w{Xn} = +00.

Proposition 2.8. 1f sUPn W{Xn} = +00, then, for every neighborhood V of e in X, one has infn,xEV IXn(x)1 = O.

Proof. Assume the contrary and fix some neighborhood V of e in X for which infn,xEv IXn(x)1 = a > O. Aside from a negligible set in V, the series Ln 1</J(Xn)l2IXn(x)l2w2 {Xn} converges. As IXn(x)1 2: a for any n, this shows that Ln 1</J(Xn)l2w2{Xn} < +00 whenever </J E f2(X, w) and this clearly implies SUPn W{Xn} < 00. •

Remark 2.8.1. Proposition 2.8 is just a slight improvement of the following

immediate consequence from assumption B): as w{Xn} = Ilxnll2"2,

sUPn ro{Xn} = +00 implies that limn---+ CXl IIXnl12 0, so that limn->CXl Xn(x) = 0 almost everywhere with respect to a.

2.9. It is time now to discuss some problems: (i) Case (A) is the dosest to the group case for which ro{Xn} = 1

for any n. We can make the quest ion precise in the following way. Let G be a semisimple compact connected Lie group. Describe the

Gel'fand pairs (G, K) such that the hypergroup M (K\ G / K) falls into case (A).

(ii) If the hypergroup (M(X), *, V) is such that the involution is trivial then its characters are real functions. Because of the orthogonality relations, any non trivial character must have zeroes. Given a character X, if Zx is its zero set in X, is it true that a(ZxJ = 07 Let Z = Un ZXn; is Z dense in X?

Notice that in case of the hypergroup (J) associated with the Gel'fand pair (TaZ2, Z2), the following properties are easily checked: (J) falls into case (A), its characters are real and Z = X.

2.10. For the next result of this section, we assume the hypergroup to belong to the dass (A); SUPn ro{Xn} < +00.

Theorem 2.11. Let c/J E C2 (X, ro); then for almost every tE [0,1], the function exp{l-llftI 2} E L1 (X, a) for I-l E R. In particular !t E LP(X, a) for any p < 00, for almost every t E [0,1].

Proof. If the hypergroup belongs to dass (A), then

n

So Ln 1c/J(XnWIXn(x)1 2ro(Xn) :::; MLn 1c/J(XnWro2 {Xn} for every X E X; by the dassical result of [Z,5.5.1.(ii)] we know that by letting

n

where the right hand series converges almost everywhere in

[0,1], one has Jo1 exp f.l1(tWdt :::; 2:%:0 t~ (f.lC)k < +00 where C = 2: 1c/J(XnWro2{Xn}, for some sufficiently small f.l > O. Therefore,

Jo1 eXPf.llft(xWdt :::; 2:%:0 t~ (f.lC)k < +00 for every x E X, and the

same f.l > O. Finally Jx Jo1 eXPf.lI!t(xWdtda(x) < +00.

42 M. O. Gebuhrer

This proves the claim for sufficiently small J-t > 0 by Fubini's theorem. Now let Sn(t; x) := :Lp::;n r p(t)4>(Xp)Xp(x)w{Xp}. If J-t is now arbitrary, the same result applies to 1ft (x) - Sn (t, x W provided n is large enough (depending on J-t) for almost every t E [0, 1].

FinaIly, using again [Z, 5.5.1 (ii)], we see that

and fixing n large enough, Sn (t, .) is bounded, so the claim is completely proved along the classical lines. •

2.11.1. Remarks. (i) This is about everything that can be extended along classical

lines in an easy way. Let C(X) be the space of continuous functions on X; it is of interest to decide whether if E is a Sidon set of the hypergroup X, one always has the property F(C(X))IE = €2(E, w). The problem whether the condition sUPn w{Xn} < +00 is sufficient, necessary or both will be addressed in the forthcoming [Cl in full detail. Related results are to be found in [vh as weIl as in [V]4, [L]. However, these papers do not lead to a conclusive statement relative to the above question.

(ii) Theorem 2.11 can also be inferred from [F - R].

3. Multipliers

In this section we just prove some easy extensions of weIl known fesults on a compact commutative group. Their mere interest, if any, is the crucial use of 1.4, which already is fairly deep.

Definition 3.1. Let X be a compact commutative hypergroup. If 4> is a complex valued function on X, we say that if F, Gare two sets of objects living on X (functions, measures,oo), 4> is a multiplier of type (F, G) if 4>.] E FC for any f E F. The set of such multipliers is denoted (F, G) as usual.

Let us first mention the following theorem.

Theorem 3.2.

(i) (C, LOO ) = (LOO , LOO ) = FM where C denotes the space C(X), and M denotes the space M (X) .

(ii) (M, M) = FM.

Proof. The proof follows the classical lines used already in the compact abelian group case. One uses 1.4 in an essential way. The result,

along with more general multiplier theorems, already appeared in Wh . •

3.3. Finally we explore the following question. Which complex valued functions cp on X have the property that the series Ln CP(Xn)!(Xn)xnW{Xn} has uniformly bounded partial sums for each continuous function f on X?

3.4. Definition and Proposition Let

Lb' := {g E LOO(X, a); sup 11 L §(Xn)xnw{Xn}lloo < +oo}. N n<;'N

We let Ilglloo,~ = sUPN 11 Ln<N §(Xn)Xnw(Xn)lloo for gELb'. Then Lb' is a Banach space and 11.illoo :::; Ilglloo,~.

Proof. First of all it is clear that Ilglloo,~ is a norm on Lb'. We prove that Ilglloo :::; IIglloo,~ for gELb'. In fact let gELb' and

define gN := Ln<N §(Xn)xnw{Xn}. The sequence (gN) is bounded in LOO(X, a) hence w* relatively compact. Ifr is a w*-accumulation point of (gN), it is immediate that t(Xn) = §(Xn) for all n, and hence r = g in Loo .

Moreover IIglloo = 1Ir!l00 = sUPllflh91 J r fdal but Jx r fda = 1imk J gNkfda so that

11 r fdal :::; s~p IlgNk 11 :::; IIglloo,~ for any f E L1(X, a), Ilflh :::; l.

Finally Ilglloo :::; Ilgll, Ilglloo,~ as claimed. Now, if (gk) is a Cauchy sequence in Lb', (gk) is also Cauchy in

Loo by the former inequality. Let g be its limit in Loo . Fix c > 0 and ke E N such that for k, k' > ke one has Ilgk -

gk/lloo,~ :::; c. Then, letting SN(g) = Ln<N §(Xn)xnw{Xn} for g E

LOO(X, a), one has IISN(gk)-SN(gk l )1100 :::; c-for any N. Letting k' go to infinity and noticing that limk' SN (gk l ) = g for any N, in LOO(X, a) one has IISN(gk) - SN(g)lloo :::; c for all N and k 2: ke . Hence g = lim gk

k-oo in Lb' and the Proposition is proved. •

Theorem 3.5. The multipliers (C, Lb') are precisely those of the form cP = P, where J1 in M(X) satisfies

44 M. O. Gebuhrer

Moreover, the function cP has the property that Ln CP(Xn)](Xn)xnw{Xn} has uniformly bounded partial sums for each continuous function f on X if and only if cP E (C, L'{') and in that case

(i) the series Ln </>(Xn)}(Xn)Xnw{Xn} is uniformly convergent for each continuous function fand

(ii) the series Ln </>(Xn)](Xn)xnw{Xn} converges in the LP norm for every f E LP(X, 0-) for 1 ~ p < 00.

Proof. We follow the lines of ([E], p. 258) where virtually anything can be retained, aside from the essential use of 1.4. See below.

First of all if cP = P and sUPN 11 Ln<N P(Xn)xnw{Xn}lll < +00, then for f E (CX), one has: -

sup 11 L P(Xn)](Xn)xnw{Xn}lloo n n'5:.N

SN(J.,,) = L P(Xn)](Xn)xnw{Xn}. n<;;'N

So SUPN 11 Ln<N P(Xn)](Xn)xnw{Xn}lloo ~ mllflloo. Therefore the mapping f 1-+ "F- 1 (cp]) is continuous from C into L,{" or cP E (C, L'{').

Conversely, if cP E (C, L,{,), the mapping f 1-+ ;:-1 (cp]) is

continuous by the closed graph theorem; let U</> '(f) = cp j. Then sUPN 11 SN (U</>(f)) 1100 ~ m· IIflloo for some constant m. Let SN(cp) = Ln<N CP(Xn)xnw{Xn}; then SN(U</>(f)) = SN(cp) * fand the former inequality showsthat sUPN 11 SN (cp) 11 1 < +00.

Therefore, the sequence (SN(CP))N is bounded in M(X), and hence there exists a w*-accumulation point /-L E M(X) which is readily seen to satisfy cP = P and sUPN IISN(/-L)lh < +00.

Now, if f E C(X), there is "I E P(X) such that IIf -"11100 ~ E. This follows from 1.4 again. (As P(X) is not always an algebra for pointwise multiplication, the Stone Weierstrass theorem cannot be used here.)

For any N, we have

where m = SUPN IISN(tI)lh· Now, /1 * "f E P(X) and SN (/1 * "f) = /1 * "f for N 2 Nah)· Therefore IISN(/1 * 1) - SN' (/1 * 1)1100 ~ Ern if N, N' 2 Nah). Thus the uniform convergence of the series is proved. The last assertion is trivial, using again 1.4 in order to get Ilf - "flip ~ E for 1 ~ p < 00 for f E LP (X, er). •

3.6. Remark. In contrast with 3.2, it is rather unexpected that 3.5 will hold in

general.

Acknowledgments. The author thanks gratefully the referee for pointing out various important references, and making some useful remarks concerning redaction and style, as weIl as for numerous interesting mathematical remarks.

References

[B-H] W.R. BLOOM, H. HEYER, Harmonie analysis of probability measures on hypergroups, De Gruyter Studies in Mathematics 20 (1995), De Gruyter; Berlin, New-York.

[E] R.E. EDWARDS, Fourier Series, A modem Introduetion Vol. II, Holt, Rinehart and Winston, Inc. (1967).

[F] J.J.F. FOURNIER, K.A. ROSS, Random Fourier series on eompaet abelian hypergroups, J. Australian Math. Soc. 37 (1984), 45-81.

[G] M.O. GEBUHRER, Ghanging signs in a Fourier series on a eompaet eommutative hypergroup, in preparation.

[Gl ] M.O. GEBUHRER, Bounded measure algebms: a fixed point theorem approach. In "Applieations of hypergroups and related measure algebms" (Providenee, RI) (0. Gebuhrer, W.G. Gonnett, A.L. Schwartz, eds.) A.M.S., Contemporary Mathematics 183 (1995), 171-190.

[G2] M.O. GEBUHRER, Analyse Harmonique sur les espaees de Gel'fand-Levitan et applieations a La theorie des semigroupes de eonvolution, These de Doctomt d'Etat (1989 Universite Louis Pasteur Strasbourg).

[G-Scl]M.O. GEBUHRER, A.L. SCHWARTZ, Bidon sets and Riesz sets for some measure algebms on the disko To appear in Colloquium Mathematicum.

46 M. O. Gebuhrer

[G-Sc2]M.O. GEBUHRER, A.L. SCHWARTZ, Sidon sets on eompaet eommutative hypergroups, in preparation.

[H] S.HELGASON-, Groups and Geometrie Analysis. (Integral Geometry, Invariant Differential Operators and Spherieal funetions), Academic Press (1984).

[L] LASSER, Laeunarity with respeet to Orthogonal Polynomial Sequenees, Acta. Seien. Math. 47 (1984), 391-403.

[S] R. SPECTOR, Apen;u de la theorie des hypergroupes. In "Analyse Harmonique sur les groupes de Lie" Seminaire NaneyStrasbourg, Lecture Notes in Mathematics 497 Springer (1973-75), 643-673.

[VI] R. VREM, Harmonie Analysis on Compaet Hypergroups, Pac. J. Math. 85 (1979), 239-25l.

[V2] R. VREM, Thesis.

[V 3] R. VREM, Laeunarity on Compaet Hypergroups, Math. Z. 164 (1978), 93-104.

[V4 ] R. VREM, Independent Sets and Laeunarity for Hypergroup, J. Austr. Math. Soc. (Series B) 50 (1991), 171-188.

[Z] A. ZYGMUND, Trigonometrieal series, Dover Publications S 290 (First edition in (1935; references are W.r.t. edition of 1955)).

Email: [email protected]; Institut de Recherche Mathematique Avancee, Universite Louis Pasteur et C.N.R.S. 7, rue Rene-Descartes, 67084 Strasbourg Cedex

Disintegration of Measures

Henry Helson

Let (X, B), (Y, C), be measurable spaces and let J-L be a measure on the product measure space. A disintegration of J-L is a representation

(1)

where for each Y in Y, v y is a measure on X. The meaning of this formula is that for each cjJ in some specified dass of function on X x Y,

(2)

(The definition is not symmetrie in the variables; if J-L has such a disintegration, it might or might not have a disintegration

J-L = J TJxdp(x) .) (3)

The first theorem of this kind was stated by J.L. Doob, although the statement and proof were incomplete. A complete and complicated discussion of the problem is contained in [6]; the interest for statistics of the problem is shown in [5]. An accessible theorem about disintegration is given in [3, p. 318], and ascribed to Abrahamse and Kriete [1]. Bourbaki has another version [2]. Some count ability hypothesis always appears. The purpose of this note is to give a theorem that is quite general and whose proof is simple.

Such a result does not belong to pure measure theory. It seems necessary to ass urne that X and Y are topological spaces of particular kinds, and then the natural statement is that (2) holds for bounded continuous functions cjJ.

Theorem 1. Let T be the unit circle, Y a compact Hausdorff space, and J-L a probability measure on the Baire field of T x Y. There is a probability measure 'Y on Y, and for alm ost every y in Y (with respect to 'Y) a probability measure v y on T such that (2) holds for all continuous functions cjJ on the product space.

If (2) has been shown to hold for all continuous functions cjJ that are products 'ljJ(X)TJ(Y) , then the Stone-Weierstrass theorem shows that

48 H. Helson

the formula holds for all continuous cjJ. Furthermore, it is enough to prove (2) when 'l/J is an exponential enix . That is, it will suffice to find measures "( and v y such that

(4)

for all integers n and all continuous functions "1 on Y. For each integer n, let (n be the complex measure on Y such that

(5)

for continuous functions "1. (The right side is obviously a continuous linear functional of "1.) Let "( be a probability measure on Y such that each (n is absolutely continuous with respect to ,,(, and write den and"(, where an is a function summable for "(.

Lemma. For a.e. ("() y, (an(y)) is a positive definite sequence.

Let (cn ) be any complex sequence with finite support. We shall show that

(6)

is a positive measure. Then 2: crcsar-s(Y) 2 0 a.e. ("(). The same inequality will be true at once for all sequences (cn ) whose values are rational numbers, except for y in a countable union of null sets. Hence the inequality will be true for all sequences without restriction except on the same null set, and this will prove the lemma.

Let "1 be any continuous non-negative function on Y. We have

J "1(y) L crcsar-s(y)d"((y) = J "1(y) L crcsd(r-s(Y)

= L crcs J J e-(r-s)ix"1 (y)d/1(x, y) (7)

= J JIL cne- nix l\(y)d/1(X, y) 20.

This proves the assertion, and the lemma. By the theorem of Herglotz, there is for a.e. ("() y a positive measure

Disintegration of Measures 49

V y on T such that vy (n) = Cl:n (y) for each n. Then

J J enixTJ(y)dJ-l(x,y) = J TJd(n = J TJCl:nd, = J TJ(y)vy(n)d,(y) ,

(8) which is the formula (4) that was to be proved.

The measure , was chosen to be a prob ability measure, but the v y

are only known to be positive. The total mass of vy is vy(O) = Cl:o(Y). Taking n = 0 and TJ the constant function 1 in (8) shows that Cl:o(y)d,(y) is also a prob ability measure, so we replace each v y by v y / CI: 0 (y) to obtain a probability measure, and , by Cl:o(y)d,(y). This completes the proof.

(Actually "( is the projection of J-l of the space Y: For each Baire set of E of Y, "((E) = J-l(T xE). It could have been defined this way, but that would have complicated the proof.)

In order to make the methods of harmonie analysis applicable, the first factor T had to be a group. We shall extend the theorem by using the fact that every uncountable standard Borel space is Borelisomorphie to T.

Theorem 2. Let X be a standard Borel space and Y a compact Hausdorff space. Give X x Y the Borel structure that is the product of the Borel field of X and the Baire field of Y. Then every probability measure J-l on X x Y has a disintegration, and (2) holds for alt bounded measurable functions cp.

First we show that formula (2), which has been shown to hold for continuous functions cp on T x Y, is actually true for all bounded Baire functions. We need the techniques of [4, Sections 50, 51]. Let S be the set of all bounded real Baire functions cp on T x Y for which the inner integral on the right side of (2) is a measurable function of y, and the formula holds. The dass S is dosed under bounded pointwise convergence, by the dominated convergence theorem. Therefore it contains the indicator function of each compact Go set. Furthermore, the sets whose indicator functions are in S form a monotone dass. The dass So of all finite, disjoint unions of proper differences on compact Go sets forms a field [4, Theorem 51.F], and the indicator functions of such sets belong to S. The smallest monotone dass containing a field is a a-field, so S contains the a-field generated by the compact Go sets, whieh is the Baire field. Hence (2) is true if cp is the indicator function of a Baire set, and therefore holds for any bounded Baire function.

If X is a countable set, the theorem can easily be proved directly. Let X be an uncountable standard Borel space, and let () be a mapping

50 H. Helson

of X onto T that is a Borel isomorphism [7]. The Baire field T of T x Y is the product of the Borel field of T and the Baire field of Y. Form the product X of the Borel field of X with the Baire field of Y. Then the mapping () in duces a Borel isomorphism of X x Y and T x Y, and the statement of the theorem is obvious.

If X is a compact metne space, its Borel field makes it a standard Borel space, so that Theorem 2 holds. In this case X is separable, but Y need not be.

References

[1] Abrahamse, M.B., and Kriete, T.L., The spectral multiplicity of a multiplication operator, Indiana Math. J. 22 (1973), 845-847.

[2] Bourbaki, N., Elements de mathematiques, Livre VI, Hermann et Cie, 1959.

[3] Conway, J.B., The Theory of Subnormal Operators, Amer. Math. Soc., 1991.

[4] Halmos, P.R., Measure Theory, Springer-Verlag, 1974. [5] Le Cam, L., Asymptotie Methods in Statistieal Deeision Theory,

Springer-Verlag, 1986. [6] Pachl, J.K., Disintegration and compact measures, Math. Seand.

43 (1978), 157-168. [7] Parthasarathy, K.R., Probability Measures on Metrie Spaees, Aca

demic Press, 1967.

Department of Mathematics, University of California, Berkeley

Multipliers of de Branges-Rovnyak spaces

11

Benjamin A. Lotto*and Donald Sarason

Abstract

Given a nonextreme point b of the unit ball of Hoc, the multipliers of the de Branges-Rovnyak space 1i(b) lie in an auxiliary space M (ä) n Hoc, where a is a function in Hoc that is associated with band M(ä) is the range ofthe Toeplitz operator Tä on H 2•

An example is constructed here to show that M(ä) n Hoc need not be an algebra. This contrasts with the case where b is an extreme point, where the analogous auxiliary space is always an algebra.

For 1 ::; p ::; 00, let V denote the usual Lebesgue space of functions on the unit circle 8D and let HP denote the classical Hardy space, thought of either as a subspace of V or as aspace of holomorphic functions on the unit disk D. For cP in Loc, the Toeplitz operator with symbol cP is the operator T", defined by the formula T",f = P+(cPf) , where P+ denotes the orthogonal projection from L2 onto H 2 .

Given a function b in the unit ball of Hoc, we define 'H.(b) , the de Branges-Rovnyak space with symbol b, to be the range of the operator (1 - nTij)1/2 on H 2 • We give 'H.(b) the range norm

where f..lker(l- nTij)1/2. This norm makes 'H.(b) into a Hilbert space. The basic properties of de Branges-Rovnyak spaces can be found in [6].

We are concerned with the multipliers of 'H.(b) , that is, the functions m in Hoc for which mf belongs to 'H.(b) whenever f does. Multipliers of 'H.(b) are studied in [3, 4]. The theory bifurcates according to whether b is or is not an extreme point of the unit ball of Hoc. In the extreme point case [3], the multipliers of 'H.(b) belong to an auxiliary space called Koc (p), which turns out to be an algebra of bounded holomorphic functions on D. It is the purpose of this note to show that the situation is different in the nonextreme point case.

·First author partially supported by NSF grant DMS-95029S3

52 B. Lotto and D. Sarason

When b is in the unit ball of Hoo but not an extreme point, a theorem of K. de Leeuw and W. Rudin [1] tells us that log(l - Ib1 2) is integrable over 3D, so there is an HOO function a such that lal 2 = 1 -lW on D [5, Theorem 17.16, p. 343]. We take a to be out er and positive at the origin; this defines a uniquely. The space M(ä) is defined to be the range of Ta on H2; as with H(b), we make M(ä) into a Hilbert space by giving it the range norm

IITa11IM(a) = 111112.

(The kernel of Ta is trivial so we don't need to include the extra condition that 1..L ker Ta.) It turns out that every multiplier of H(b) belongs to M(ä) and that the space M(ä) n HOO is the auxiliary space analogous to the space KOO (p) mentioned above. The quest ion thus arises whether M(ä)nHOO is an algebra. We shall show that it need not be. Specifically, we construct a nonextreme point b such that bitself belongs to M(ä) but b2 does not. The property that we will use to identify whether or not functions belong to M(ä) is given in the following definition.

Definition 1. Suppose that 1 and 9 belong to Hoo. We say that (1, g) is an H2-corona pair i1 there exist 1unctions <p and'lj; belonging to H2 with 1<p + g'lj; = 1.

K. C. Lin [2] has found both necessary and sufficient conditions on 1 and 9 for the pair (1, g) to be an H 2-corona pair. These conditions involve an estimate of the nontangential maximal function of 1/(1112 + IgI 2)1/2. Unfortunately, there is a gap between the two conditions. It would be nice to have simple necessary and sufficient conditions on 1 and 9 for (1, g) to be an H 2-corona pair.

Our first result connects being an H 2-corona pair with membership in M(ä). We write H'5 for the space of functions in H2 that vanish at the origin and H2 (respectively Hg) for the space of complex conjugates of functions in H2 (respectively H'5). Note that (H2)1. = Hg.

Theorem 1. Suppose that 1 is in Hoo and that (c -1112)/lal belongs to Loo for some positive constant c. Then 1 belongs to M(ä) i1 and only if (1, a) is an H2 -corona pair.

Proof. First, suppose that 1 belongs to M(ä) and write 1 = Tag with 9 E H2. Then 1 - äg belongs to Hg and so J - ag belongs to H'5. Denote this function by <p. Multiplying by 1 keeps us in H'5, so we find that

Mulipliers of de Branges-Rovnyak Spaces II 53

belongs to Hg. Hence the second term on the right belongs to H2. Since a is out er and the quantity in brackets belongs to L 2 , we can divide by a and conclude that the quantity in brackets actually belongs to H 2 • Denote this quantity by 'ljJ. Then cP and 'ljJ belong to H 2 and fcP + a'ljJ = Ifl2 - afg + (c -lfI2) + ajg = c. Dividing cP and 'ljJ by c, we see that (1, a) is an H2-corona pair.

For the other direction, suppose that (1, a) is an H 2-corona pair. Let cP and 'ljJ be H 2 functions that satisfy fcP + a'ljJ = 1. Then !~ + äij; = 1, so

f If12~ + äfij;

ä [- c - dfl2 ~ + fij;] + c~. Let p denote the term in brackets. Then p belongs to L 2. Applying P+ to the equality above we get

f = P+(äp) +ccP(ü) = Ta (P+p + ccP(ü)/a(ü)) ,

so f belongs to M(ä) as desired. • Remark 1. The first part of the proof goes through as written under the weaker hypothesis that (c - IfI2)/lal E L 2 .

Remark 2. The second part of the proof go es through essentially as written under the weaker hypothesis that (c - /f/ 2 )//a/ E L 2 if we also assume the stronger hypothesis that (1, a) is a corona pair. (This means that (1, a) satisfies the conclusion of Carleson's Corona Theorem, that is, there exist functions cP and 'ljJ in Hoo such that f cP + a'ljJ = 1.)

Remark 3. If we keep the hypothesis that (c - /f/ 2 )//a/ E Loo and assume that (1, a) is a corona pair, we can obtain the stronger conclusion that f is actually a multiplier of M(ä). To see this, let 9 be in M(ä) and write 9 = Tah for some h in H 2. Arguing as above, we find that

fg Ifl2g~ + äfgij;

= ä [Ch~ + c - ~f12 g~ + f9ij;] + c(g - äh)~.

The second term on the right belongs to Hg and the term in brackets (which we'll call p as above) is in L 2. It follows as before that fg =

Ta(P+p), so that fg belongs to M(ä).

Our next result is a necessary condition for a pair of functions to be an H 2-corona pair.

Lemma 1. Suppose that (f, g) is an H2-corona pair. Then there is a constant C > 0 such that

for alt z in D.

Proof. Clearly (f, g) is an H 2-corona pair if and only if the function 1 is in the range of the operator T( cf; EB 'ljJ) = f cf; + g'ljJ on H 2 EB H 2 . By a well-known condition, this happens if and only if there is a constant C > 0 such that

for all h in H 2. Now T*h = TfhEBTgh, so taking h = kz , the reproducing kernel function for H 2 at z in D, we obtain

as desired. • Remark 4. We may replace 1 - Izl2 by 1 - Izl in the above result because the ratio of the two terms is bounded above and below on D.

We are now ready to construct our example. Fix 0:, ß > 0 (to be determined later) and define the function w( ()) on ßD by

{

()a if 0 < () < 1/2

w(())= (1_1()1 2ß)1/2 if-1/2<()<0;

1/ v'2 otherwise.

Clearly w belongs to LOO and log w is integrable. We therefore may define b to be the out er function in Hoo with Ib(()) 1 = w(()). It is easy to check that 10g(1 - IW) is integrable, so by the previously mentioned theorem of de Leeuw and Rudin, b is not an extreme point of the unit ball of Hoo. The corresponding function a satisfies

{ (1 - ()2af / 2 if 0< () < 1/2

la(e)1 = lelß if -1/2< e < 0; 1/ v'2 otherwise.

We need to make an estimate on b.

Lemma 2. For every E > 0, Ib(r)I/(l- r)%-€ --t 0 as r --t 1-.

Mulipliers 01 de Branges-Rovnyak Spaces II 55

Proof. Let Pz(B) denote the Poisson kernel for the point z in D. For BQ < 1/2 and 0 < r < 1 we have

log Ib(r)1

Substituting in

P'(B)- 1-r2

r - 1 + r2 - 2rcos(B)

and integrating gives

0: log BQ (1 + r BQ) log Ib(r)1 ::::; arctan -- tan - . 7r 1-r 2

Fix 0 > 0 and take BQ = (1 - r)h5. As r -+ 1-, then, we have

arctan -- tan - -+ -. ( 1 + r BQ) 7r 1- r 2 2

Taking r elose enough to 1 gives us

< 0: 10g(1",.- r)l-C (_7r2

_ 5:) log Ib(r)1 " u

(1 - 0) (1 - ~) ~ 10g(1 - r)

< (~-E)log(l-r),

when 0 is chosen small enough so that (1 - 0)(1 - 20/7r)0:/2 ~ 0:/2 -E. Hence Ib(r)I/(l - r)~-€ is bounded above. But 10 was arbitrary, so Ib(r)I/(l- r)~-€' is bounded above for a slightly sm aller 10'. This implies that Ib(r)I/(l- r)~-€ actually tends to 0 as r -+ 1-. •

Remark 5. The same estimate applies to a and gives la(r)I/(l -ß

r)2-€ -+ 0 as r -+ 1- for any 10 > O.

Remark 6. Similar reasoning shows that for any 10 > 0, Ib(z)I/(l -Izl)"-€ -+ 0 as z -+ 1 along some line segment in D that termin at es at 1. Let rjJ be the angle that the line makes with a verticalline, measured

1. Let <p be the angle that the line makes with a vertical line, measured counterclockwise (so the case of the previous lemma is <p = 7r/2). Write z = reip for the point on the segment at distance r from the origin. Following the reasoning in the proof of the lemma, we find that

0: log 00 [ (1 + r )] (}=(}o-p

7r arctan 1 _ r tan( 0) (}=_p

Take r -t 1 and let 00 = (1 - r)l-O as before. p ~ (1 - r) / tan <p gives that the arctangent term arctan (2 cot <p), so we find that

log Ib(z)1 :::; (1 - r)ca-,

where c = 1/2 + arctan(2 cot <p) /7r.

Theorem 2. Let band a be as above.

1 1"f bn th 1-lfI2 .. Loo • J = , en lai zs zn .

2. 1f 0: < 1/2, then b E M(a).

3. 1f no: > 1 and ß > 1, then bn 1- M(a).

The estimate tends to 7r /2 +

4. 1f 0: < 1/2, no: > 1, and ß > 1, then M(a) nHoo is not an algebra.

Proof. For part 1, note that

since 1 -lW = lal 2• This belongs to Loo since both a and b do. For part 2, note that if 0: < 1/2, then l/lbl is in L2. Since b is outer,

it follows that l/b is in H 2 . Hence (b, a) is an H 2 corona pair via the identity b· (l/b) + a . 0 = 1. It follows from part 1 and Theorem 1 that bis in M(a).

Now part 3. It follows from Lemma 2 that Ibn (r)1 2/(1 - r) -t 0 as r -t 1-. Similarly, from the first remark following Lemma 2, a(rW/(lr) -t 0 as r -t 1-. It now follows from Lemma 1 that (bn , a) is not a

Multipliers 01 de Branges-Rovnyak Spaces II 57

eorona pair. Henee from part 1 and Theorem 1, bn does not belong to M(a).

Finally, part 4 follows immediately from parts 2 and 3. •

Remark 7. The hypotheses for part 4 of Theorem 2 ean be aehieved by taking a to be slightly less than 1/2, n = 3, and ß to be anything larger than 1. This gives an example of a function b for whieh b is in M(a) but b3 is not in M(a).

Remark 8. In order to get the promised example (b in M(a) but b2 not in M(a)), we have to modify the previous theorem slightly, making the estimates of band a along a segment in D that terminates at 1 different from the radius. We use the estimate from Remark as weH as the notation introdueed there. Let z = re ip --+ 1 along a very nearly vertieal segment. Then Ib(re ip ) 1/(1- r)a-f --+ 0 as r --+ 1-. If we take a to be less than 1/2, we find that l/b is in H 2 as above so that (b, a) is an H 2-eorona pair and so b is in M(a). Now Ib2 (re iPW/(1- r)4(a-f) --+ 0 as r --+ 1-. Choose a > 1/4 and E > 0 sm all enough so that 4(a - E) > 1. (These ehoiees will dietate the exact segment along whieh we make our estimate.) This being done, take ß large enough so that la(reiPW /(1 - r) --+ 0 as r --+ 1-. We ean now show as above that (b2 , a) is not an H 2-eorona pair, so that b2 is not in M(a).

As a final remark, we note we have a faetor of two to play with in this example (we ean ehoose 1/4 < a < 1/2). The gap between Lin's neeessary and sufficient eonditions for a pair of functions to be an H 2_

eorona pair also involves a factor of two. We are eurious if there is any signifieanee to this.

Acknowledgements. The first author would like to thank Dinesh Singh and the University of Delhi for their hospitality during the Seeond International Conferenee on Harmonie Analysis.

References

[1] K. de Leeuw and W. Rudin, Extreme point and extremum problems in H 1, Paeijie J. Math., 8 (1958), 467-485.

[2] Kai-Ching Lin, On the HP solutions to the eorona problem, Bult. Sei. Math., 118 (1994), 271-286.

[3] B. A. Lotto and D. Sarason, Multiplieative strueture of de Branges's spaees, Revista Matematica Iberoamericana, 7 (1991), 183-220.


[4] B. A. Lotto and D. Sarason, Multipliers of de Branges-Rovnyak spaces, Indiana Univ. Math. 1., 42 (1993), 907-920.

[5] W. Rudin, Real And Complex Analysis, McGraw-Hill, New York, third edition, 1987.

[6] D. Sarason, Sub-Hardy Spaces in the Unit Disk, volume 10 of The University of Arkansas Lecture Notes in the Mathematical Sciences, Wiley, New York, 1995.

Email: [email protected]; Department of Mathematics, Vassar College, 124 Raymond Avenue, Poughkeepsie, NY 12601

Email: [email protected], Department of Mathematics, University of California, Berkeley, CA 94720

On Hartman Uniform Distribution and Measures on Compact Spaces

R. Nair

Abstract

Given a sequence of natural numbers k (kn)~=l' we say it is Hartman uniformly distributed if

N

lim ~ L e27riknX 0, N-+oo N

n=l

for every non-integer real number x. This property of k being Hartman uniform distributed is interesting in subsequence ergodic theory because if for some p in [1,2], for a function f E LP(X, ß, J-l) we have the limit

1 N lim N L f(Tknx)

N-+oo n=l

existing almost everywhere with respect to the measure J-l, then just the ergodicity of the dynamical system (X, ß, J-l, T) implies that fT,f(X) = Ix fdJ-l, which is of course useful with regard to applications. Not every sequence for which a pointwise convergence theorem holds has this property. In this paper we use this observation to give so me results about invariant measures for continuous maps of compact metric spaces.

1. Introduction

Let (X, ß, J-l) be a prob ability space and let T : X -) X be a measurable map that is also measure preserving; that is, given A E ß, we have J-l(T- 1 A) = J-l(A) , where T- 1 Adenotes the set {x EX: Tx E A}. In this paper we say a sequence k (ki)~l is LP good universal, if for each prob ability space (X, ß, J-l), each measurable measure preserving map T of it and for all functions f E LP (X, ß, J-l)

60 R. Nair

we have the limit

1 N lim N L f(Tki X) = eT,J(x),

N-+CXl i=l

existing almost everywhere with respect to the measure /1. A number of sub-sequences of the natural numbers have been shown to be LP good universal in the last few years. For instance if cp : N --t N is a polynomial mapping the integers to themselves and (Pn)~=l is the sequence of rational primes, then the the sequences of integers (cp(n))~=l and (CP(Pn))~=l are LP good universal for P in (1,00]. For the detailed proofs see [Bo2], [NaI] and [Na2] respectively. For a set S let XS denote its characteristic function. Suppose K = (nk)k=l ~ N is a strictly increasing sequence of natural numbers. By identifying K with its characteristic function XK we may view it as a point in A = {O,I}N, the set of maps from N to {O, I}. Conversely we can also view A as the space of strictly increasing sequences of natural numbers like K. We endow A with a probability measure by viewing it as the Cartesian product A = TI~l X n where for each n (n = 1,2, ... ) we have X n = {O, I} and specify the probability 7fn on X n

by 7fn({I}) qn with 0 ::; qn ::; 1 and 7fn({O}) 1 - qn where the sequence (qn)~=l satisfies limn-+ oo qnn(loglogn)-l = 0. The desired prob ability measure on A is the corresponding product measure 7f = TI~=l 7fn- The underlying a-algebra ß is that generated by the "cylinders"

for all possible choices of i 1 , ... ,ir and ail' ... , air' Then almost every K in (A, ß, 7f) is also LP good universal for each P in (1,00] [Bol]. Suppose for areal number y we denote its integer part by [y]. Suppose g(x) is a function from [1,00) to itself whose derivative increases to infinity with its argument. Let kn denote [g(n)] (n = 1,2, ... ) and let AM denote the cardinality of {n : kn ::; M}. Suppose for a function a : [1,00) --t [1,00) increasing to infinity as its argument does, that we set

b(M) sup (G)E[a(~)'!)

I L e27riGkn I n:kn~M

Suppose also that for some decreasing function c : [1,00) --t (0,00)

On Hartman Uniform Distribution and Measures 61

there exists C > 0 such that

b(M) + A[a(M)] + M a(M) < Cc(M),

AM

and we have

00

LC(OS) < 00. s=l

The sequences k = (kn)~=1 we have just described are also LP good universal for each P in (1,00]. Sequences that satisfy this condition indude the cases where g(n) = nß where ß is non-integer and greater than one; g(n) = P(n), where P(x) = ao +a1x + ... + akxk with a1, ... ,ak are not all rational multiples of the same real number and g(n) = e(logn)t with t E (1,~) [Na4].

We say a sequence is Hartman uniformly distributed if for x not an integer

1 N . lim - L e21r>k n x = o.

N-.oo N n=1

A sequence being Hartman uniformly distributed has a couple of equivalent formulations. Firstly, a sequence is Hartman uniformly distributed if and only if it is uniformly distributed on the Bohr compactifieation of the integers. Secondly, a sequenee is Hartman uniformly distributed if and only if for eaeh irrational number a, if (y) denotes the fractional part of y, the sequenee ((kna) )~=1 is uniformly distributed modulo one and further the sequenee k itself is uniformly distributed among the residue dasses modulo m for every natural number m greater than or equal to two. For more on Hartman uniform distribution see [H] or [KN]. See also [N] for more on uniformly distributed sequenees of integers and [Na3] for a large dass of sequences of integers which are Hartman uniformly distributed, not all of whieh however are known to be LP good universal. For example, the sequences (4)(n))~=1 and (4)(Pn) )~=1' mentioned earlier, are not in general Hartman uniformly distributed but the other examples mentioned above are. Recall that we say (X, ß, /1, T) is ergodic if T-1 A = A for any A E ß implies /1(A) is either zero or one. Hartman uniformly distributed sequences are interesting in subsequence ergo die theory because for them ergodicity of the underlying dynamieal system (X, ß, /1, T) implies that the limit of the ergo die average is equal to the the integral of the function

62 R. Nair

almost everywhere. This is of course very useful with regard to applications. This is not the case for all sequences (cf. [AN]). In Section 2 we describe some new formulations of ergodicity. In Section 3 we apply these to study weak convergence of measures on compact spaces and generalize to good universal Hartman uniformly distributed sequences results previously only known for kn = n (n = 1,2, ... ).

2. Conditions equivalent to ergodicity

We say a sequence of complex numbers (an)~=_CXl is positive definite if for any sequence (Zn)~=_CXl of complex numbers only a finite number of whose terms are non-zero

n,m

We have the following lemma due to Herglotz [Ka p. 38].

Lemma 2.1. A sequence oJ complex numbers is positive definite iJ and only iJ there exists a finite measure W on 1I' IRjZ such that

(n E Z)

Let U = UT denote the Koopman operator defined pointwise on LP(X, ß, J.l) for 1 :S p :S 00 by U(f(x)) = J(Tx). Recall that because T is measure preserving, for each function J in L 1 (X,ß,J.l) we have Ix J(Tx)dJ.l = Ix J(x)dJ.l and so in particular IIUJI12 = IIJI12 where as usual for 1 :s; p :s; 00, the notation 11.llp refers to the LP norm. This means that U is an isometry on the Hilbert space L 2 • Let (.,.) denote the standard inner product on L 2(X, ß, J.l). If we denote the adjoint of U by U- 1, then the sequence ((Un J, 1) )~=-CXl is positive definite; hence by Lemma 2.1 there exists a measure wJ satisfying

(n E Z)

The measure wJ is called the spectral measure.

Throughout this section we ass urne that k = (kn)~=l is LP good universal for a fixed p E [1,2]. We have the following lemma.

Lemma 2.2. If for p in [1,2] and k = (kn)~=1 which is Hartman uniformly distributed we have

in LP norm, then fT,J(Tx) = fT,J(X) /-L almost everywhere. In consequence, if (X, ß, /-L, T) is ergodic then fT,J(X) = Ix fd/-L /-L almost everywhere.

Proof. First assume that p = 2. Then

_1 ~ r (f 0 Tki +1 - f 0 Tki) N2 L.J Jx

O<ki ,kj'5.N

x (foTkj+1 - foTkj)(x)d/-L(x)

~2 l: ((Uki +1-(kj+1) f, f) - (Uki+1- kj f, f) O<ki ,kj '5. N

where z = eiO. This tends to zero with N because k = (kn)~1 is Hartman uniformly distributed. The Lemma follows for general p in (1,2] as L2 is dense in LP. •

We now come to the following central result, which together with its proof in the case where kn = n (n = 1, 2, ... ) is already known [W].

Theorem 2.3. Consider the following statements:

(i) the dynamical system (X, ß, /-L, T) is such that for each pair of

64 R. Nair

elements A and B in ß we have

1 N lim N L J-l(A n T-kn B)

N-+oo n=l

(ii) for each pair f, 9 of functions in L 2 (X, ß, J-l) we have

1 N lim N"(Ukif,g) = (f,l)(l,g);

N-+oo ~ i=l

(iii) if I is in L 2 (X, ß, J-l), Ix IdJ-l = 0 and Wf is the spectral measure defined above by the sequence ((unI, 1) )nEZ, then

(iv) if I is in LP(X, ß, J-l) for p in [1,2] then

in LP(X, ß, J-l) norm;

and

(v) il I is in LP(X, ß, J-l) for p in [1,2] then

almost everywhere with respect to J-l.

(vi) T is ergodie. Then, irrespective 01 whether (kn)~l is Hartman uniformly dis

tributed or not, (i) is equivalent to (ii), and (iii) is equivalent to (iv). Also (ii) implies (vi), and (v) implies (i). Further in light of Lemma 2.2 (vi) implies (iv) and (v), so assuming (kn)~=l is Hartman uniform distributed we see that all the conditions are equivalent.

Proof. That Theorem 2.3 (ii) implies Theorem 2.3 (i) is manifest because (i) is the special case of (ii) where I = XA and 9 = XB·


Proof that Theorem 2.3 (ii) implies Theorem 2.3 (i). In the special case where for two sets A and B in ß we set a = XA and b = XB (i) may be rewritten

(2.1) (a, 1) (1, b).

Also, by taking linear combinations of characteristic functions, this statement is also seen to remain true for simple functions a and b. For two arbitrary L2 (X, ß, J.l) functions fand 9 given E > 0 we can find simple functions a and b such that Ilf - al12 :::; E and 119 - bl1 2 :::; E.

Further by (2.1) we can find a possibly large natural number n = n(E) such that if N 2: n(E) then

Thus, also if N > n(E)

1 N 1 N

+ INL(Ukia,g) - NL(Ukia,b)1 i=l i=l

N

+ IL(Ukia,b) - (a,l)(l,b)1 i=l

+1(a,l)(l,b)

+ IU,l)(l,b)

(1,1) (1, b) I

(I,l)(l,b)1

66 R. Nair

which is

+ E + IU- a,l)(l,b)1 + IU,1)(1,9-b)l.

Using the fact that IIfl12 = U, f) ~ and Cauchy's inequality this is

thereby completing the proof that Theorem 2.3 (ii) implies Theorem 2.3 (i), as required. •

Proof that Theorem 2.3 (ii) implies Theorem 2.3 (iv). If T- 1 A = A then (ii) implies that j,L(A) = j,L2(A), and so j,L(A) is either zero or one and T is ergodic. In light of Lemma 2.2 this proves Theorem 2.3 (iv) .

• Proof that Theorem 2.3 (iv) implies Theorem 2.3 (v). Let (Nt)~1 denote a strictly increasing sequence of integers such that

The existence of such a sequence is ensured by the assumption of Theorem 2.3 (iv). As a consequence we have

Rearranging, a step justified by Fatou's lemma, we have

We may therefore conclude, using the fact that for a sequence (at)~1 of non-negative functions 2:::1 at < 00 implies that at = 0(1), that


we have

1 Nt 1 lim - '" f(Tki X ) = X f(x)djL(x),

t __ co Nt ~ i=l

almost everywhere with respect to jL. This means that CT,J(X) Ix f(x)djL jL almost everywhere as required. •

Proof that Theorem 2.3 (v) implies Theorem 2.3 (i). Note that by (v) for any two sets A and B be10nging to ß we have

almost everywhere with respect to jL. This me ans that as a consequence of the dominated convergence theorem (i) follows. •

Proofthat Theorem 2.3 (iii) and Theorem 2.3 (iv) are equivalent. Again without 10ss of generality we assume that Ix f(x)djL = o. The proof of the equiva1ence of (iii) and (iv) now follows from the following identity.

which via the properties of the spectra1 measure is

This proves that (iii) and (iv) are equiva1ent if p = follows as before because L2 is dense in LP.

2. The case p ~ 2

• Proof that Theorem 2.3 (v) implies Theorem 2.3 (iv).

Suppose first that 9 is bounded and measurab1e; then in particu1ar 9 is in LP and so assuming (v) we have

68 R. Nair

almost everwhere with respect to J1. This means that by the bounded convergence theorem we have

Now if we are given f > 0 then there exists a possibly large natural number n n(f,g) such that if N > n and k is a positive integer then

Let us now consider general functions f in LP(X, ß, J1) and set

We want to show that (SN(f))N=l is a Cauchy sequence in LP(X, ß, J1). First notice that IISN(f)llp ~ Ilfllp" Suppose we are given f > 0 and that 9 is in LOO(X,ß,J1) and that Ilf - gllp < f. Then

IISN(f) - SN+k(f)llp ~ IISN(f) - SN(g)llp

+ IISN(g) - SN+k(g)llp

+ IISN+k(g) - SN+k(f)llp ,

which is less than f, if N > n(~,g) . Thus (SN(f))N=l is a Cauchy sequence, implying that

for some function f*. But g*(x) = Ixg(x)dJ1 for gin LOO(X,ß,J1) and such functions are dense in LP(X, ß, J1) and so f*(x) = Ix f(x)dJ1 for all functions f in LP(X, ß, J1) as required. Hence Theorem 2.2 (iv) and hence the whole of Theorem 2.2 is proved as required. •

On Hartman Uniform Distribution and Measures

3. Some consequences for measures on compact metrisable spaces

69

Throughout this section X will denote a compact metrisable space and ß = ß(X) will denote its Borel a-algebra. For a measurable transformation T of the space X we denote by M(X, T) the set of T invariant probability measures on the measurable space (X, ß). Also throughout this section we ass urne k = (kn)~=l is both Hartman uniformly distributed and LP good universal. Suppose also that T is continuous. We denote by C(X) the space of continuous complexvalued functions defined on X. We then have the following theorem.

Theorem 3.1. Let J-t be an element of M(X, T). Then T is ergodic if and only if for each f in C(X) and 9 in L 1 (X,ß,J-t) we have

(3.1)

Proof. Suppose first that (3.1) holds far each f in C(X) and each 9 in Ll(X,ß,J-t), and let Fand G belong to L2 (X,ß,J-t). In particular G is in Ll(X,ß,J-t) and so

for all f in C(X). Now approximate F in L2 (X, ß, J-t) by continuous functions to get

1 N 1 1 1 lim N L F(Tknx)G(x)dJ-t(x) = X FdJ-t X GdJ-t. N-+oo n=l X

Hence (3.1) implies the ergodicity of T. We now show the converse. Suppose the transformation T is ergodic with respect to the space (X, ß, J-t) for some measure J-t in M(X, T). Suppose also that f is an element of C(X) and hence also an element of L2(X, ß, J-t). Thus if h is also an element of L2 (X, ß, J-t) we get

70 R. Nair

If 9 is in L 1 (X, ß, J-l) then by approximating 9 by elements h of L2(X, ß, J-l) we obtain

1 N 1 lim N L f(Tknx)g(x)dJ-l(x) N-.oo n=l X

as required. • Suppose (X, ß, J-l, T) is an ergodie transformation. The next ap

plication concerns measures m which are absolutely continuous with respect to J-l. For a measure m its pullback T-1m under the transformation T is defined by T-1m(A) = m(T-1 A) for all A in ß where as before T- 1 A = {x : Tx E A}.

Theorem 3.2 : Suppose the measure J-l is an element of M(X, T). Then (X, ß, J-l, T) is ergodie if and only if whenever m is in M(X, T) and m is absolutely continuous with respect to J-l then

(3.2)

weakly in M(X).

Proof. First suppose (X, ß, J-l, T) is ergodic and suppose m in M(X) is absolutely continuous with respect to J-l. Let 9 denote the RadonNikodym derivative ~; which is contained in L1(X,ß,J-l)' If f is in

C(X) then

1 1 N 1 N 1 lim fd( - L T - knm ) = lim N L f 0 Tkndm

N-.oo X N N-.oo X n=l n=l

1 N 1 lim N L f(Tknx)g(x)dJ-l N-.oo n=l X

Thus


which is (3.2) as the theorem promises. We now consider the converse. Suppose (3.2) holds. Let g be an element of LI (X, ß, /1) and first suppose that g is positive everywhere. Set m to be the measure defined on X by m(B) = c Ix d/1 where c = ~. Then if f is in C(X) we

Jx gdJ.L

have

= lim ~ t r f(T kn x)g(x)d/1 = r fd/1 r gd/1. N->oo n=1 Jx Jx Jx

Now suppose gis in LI (X, ß, /1) and real valued, write it g = g+ - g_ where g+ and g_ are the positive and negative parts of g respectively. The desired conclusion for g follows by applying the deduction of the previous paragraph to g+ and g_ in turn. The case of complex valued g is treated similarly. •

The following theorem says that in the context of continuous functions on X there is a universal convergence set.

Theorem 3.3. If /1 is in M(X, T) for the continuous map T of X and (X, ß, /1, T) is ergodic, then there exists a set Y in ß with /1(Y) = 1 such that for each f in C(X) and each y in Y we have

Proof. Choose a countable dense subset (fk)k=1 of C(X). Using Theorem 3.1 there is a set Xk in ß such that /1(Xk) = 1 and

for all x in X k. Put Y = nk:;::IXk. We have /1(Y) = 1 and

for all y in Y and k in N. The theorem now follows by approximating a given f in C(X) by members of (fk)k=l' •

72 R. Nair

The following lemma relates delta measures to ergodicity.

Theorem 3.4. Suppose T is a continuous map of X and that /-L is in M(X, T). Let Oy denote the delta measure at y; that is, for each set A in ß oy(A) = 1 if y is an element of A and oy(A) = 0 otherwise. Then (X, ß, /-L, T) is ergodie if and only if we have

1 N lim N '"' OTkn y /-L, N--+oo ~

n=l

weakly /-L almost everywhere in y.

Proof. If (X, ß, /-L, T) is ergodie then by Theorem 3.3 we immediately have

weakly in M(X) for all y in Y, where /-L(Y) = 1. Conversely suppose

1 N lim N L OTkn y = /-L,

N--+oo n=l

weakly for all y in Y with /-L(Y) = 1. Then for all f in C(X) we have

1 N 1 lim - L f(Tkny) = fd/-L, N--+oo N x

n=l

for all y in Y. If y is in Y and f is in C(X) and 9 is in L 1 (X, ß, /-L) then

1 N 1 lim N L f(Tkny)g(y) = g(y) fd/-L, N--+oo ~1 x

and so, integrating both sides over Y with respect to /-L, Theorem 3.1 yields the ergodicity of (X, ß, /-L, T) as required. •

Recall that we say that (X, ß, /-L, T) is uniquely ergodic if the only element in M(X, T) is /-L. Necessarily this means (X, ß, /-L, T) is an ergodie transformation. We need a lemma.

Lemma 3.5. The following statements are equivalent :


(a) the transformation (X, ß, /1, T) is uniquely ergodic ; (b) for each continuous function f defined on X there is a constant

Cf independent of x such that

uniformlyon X; and (c) whenever f is in C(X)

1 N lim N L f(Tknx)

N-->oo n=l

pointwise on X.

Proof. We first consider the proof that (c) implies (a). Let

(N = 1,2, ... )

For v in M(X, T), by the dominated convergence theorem we have

lim r SNf(x)dv(x) = r fdv = r fd/1. N -->00 } x } x } x

This holds for all f in C(X) and hence by the Riesz representation theorem we have v = /1, as required.

We now show how (a) implies (b). Suppose (b) does not hold. Then there exists an E > 0, a function g in C(X) and a sequence (xnj )~1 in X such that

Using the unique ergodicity property of (X, ß, /1, T) and refining (Xn)~l if necessary we can find v in M(X, T) such that

which is a contradiction. The proof that (b) implies (a) is obvious. •

74 R. Nair

We are now ready to prove the final theorem of this section. For a transformation (X, ß, /-L, T) we say a point x in X is k generic if

1 N 1 lim N L f(Tknx) = fd/-L , N-HX) n=l X

holds for aIl functions f in C(X). RecaIl that we say a continuous transformation T of X is minimal if the only closed subsets it leaves invariant are X and the empty set.

Theorem 3.6. Suppose T is minimal on X and assume for each x in X there exists a measure /-Lx such that x is k generic for (X, ß, /-L, T). Then (X, T) is uniquely ergodic.

Proof. By assumption, for every x in X

lim NI t f(Tknx) = r fd/-Lx = CU, x). N-+oo n=l Jx

We have to show that CU, x) is independent of x. Fix f in C(X). Then the function CU, x) is the pointwise limit of continuous functions and Tinvariant because of the Hartman distribution of k. Because of the assumed minimality, for Xo in X the sequence (TknXo)~=l is dense in X and so if Xo were a point of discontinuity, then CU, x) would be nowhere continuous contradicting the Baire category theorem which teIls us that the points of discontinuity of CU, x) are of first category. This 'means that CU, x) is constant as required, •

References

[AN] N. H. Asmar and R. Nair: Certain averages on the a-adic numbers, Proc. Amer. Math. Soc. 114 no. 1 (1992), 21-28.

[Bol] J. Bourgain: On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math. 61 (1988), 39-72.

[B02] J. Bourgain: Pointwise ergodie theorems for arithmetie sets, Publ. I. H. E. S. 69 (1989), 5-45.

[H] S. Hartman: Remarks on equidistribution on non-eompaet groups, Compositio Math., 16 (1964), 66-7l.

[Ka] Y. Katznelson: An introduetion to Harmonie Analysis, Wiley, (1968).


[Kr] U. Krengel: Ergodic Theorems, de Gruyter Studies in Mathematics 6 (1985).

[KN] L. Kuipers and H. Neiderreiter: Uniform Distribution of Sequences, Wiley, (1974).

[Na1] R. Nair: On polynomials in primes and J. Bourgains circle method approach to ergodic theorems, Ergododic Theory and Dynamical Systems, 11 (1991), 485-499.

[Na2] R. Nair: On polynomials in primes and J. Bourgains circle method approach to ergodic theorems II, Studia Math. 105 (3) (1993), 207-233.

[Na3] R. Nair: On uniformly distributed sequences of integers and recurrence, (preprint).

[Na4] R. Nair: On uniformly distributed sequences of integers and recurrence II, (preprint).

[N] 1. Niven: Uniform distribution of sequences of integers, Trans. Amer. Math. Soc. 98 (1961), 52-6l.

[W] P. Walters: An Introduction to Ergodic Theory, Springer-Verlag, Graduate Texts in Mathematics 79 (1981).

Department of Pure Mathematics, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, UK

Hypergroups and Signed Hypergroups

K enneth A. Ross

1. Introduction to Hypergroups

Hypergroups, as I understand them, have been around since the early 1970's when Charles Dunkl, Robert Jewett and Rene Spector independently created locally compact hypergroups with the purpose of doing standard harmonic analysis. As one would expect, there were technical differences in their definitions. The standard, in the nonSoviet world, became Jewett's 100-page paper [J] because he worked out a good deal of the basic theory that people would want. Bloom and Heyer's book [BH] is a useful report on some of the mathematics that has been done on the basis of Jewett's axioms.

In August of 1993 the first conference on hypergroups was held in Seattle. This first conference was an international conference and was weIl attended by people from all over the world. See [CGS]. As the preface to the proceedings states, "This led to fireworks. Hypergroups occur so often and in so many different and important contexts, that mathematicians all over the world have been discovering the same mathematical structure hidden in very different applications, and publishing theorems ab out these structures, in many cases without even knowing that they were talking about hypergroups."

In particular, structures very like the Dunkl-Jewett-Spector creat ions of the 1970's had been studied in the early 1950's by Berezansky and colleagues. The axioms, terminology and language were all different, and the connection was not realized by most workers until the Seattle conference in 1993. These connections are explained in the nice article [BKl] , where it is noted that the ideas of hypergroups appear in works of DeIsarte and Levitan published in 1938 and 1940. See also the very recent article [BK2]. A fundamental part of their axioms are their "structure measures" c(A, B, r) which for locally compact groups with Haar measure m reduce to m((A - r) n B). I think the axioms involving c(A, B, r), which have no counterpart in the axioms of a 10-cally compact group, are unnatural compared to the axioms of Jewett and others. In any case, in the proceedings [CGSj of the conference there was an effort to standardize notation. In many of the papers, the hypergroups studied by Jewett rand used in the book of Bloom and

78 K.A. Ross

Heyer] are referred to as DJS-hypergroups. These are the hypergroups that I will be talking about.

Incidentally, there was a follow-up conference at Oberwolfach in 1994 that was organized by Herbert Heyer. The proceedings [H] contains many interesting articles, including several on hypergroups.

Here are the axioms for a DJS-hypergroup. As I learned from [K], they have been neatly rephrased by Lasser [L], so I will give Lasser's version. We begin with a locally compact Hausdorff space K. M(K) will denote the space of all finite complex regular measures on K, and M1(K) will denote the prob ability measures in M(K). Point masses will be denoted by 8x . A hypergroup is determined by K and the following data:

(H*) A continuous mapping (x, y) ---> 8x * 8y from K x K into M1(K), where M1(K) has the weak topology with respect to the space Cc(K) of continuous complex-valued functions with compact support. [convolution]

(HV ) An involutive homeomorphism x ---> XV from K to K. [an involution]

(He) A fixed element e in K. [an identity element]

After identification of x with 8x , the mapping in (H*) has a unique extension to a continuous bilinear mapping (JL, 11) ---> JL*II from M(K) x M(K) into M(K). And the involution on K gives an involution JL ---> JL* on M(K), where JL*(E) is the conjugate of JL(EV ) for each Borel set E in K.

Now a DJS-hypergroup is the quadrupie (K, *, V, e) satisfying

(H1) 8x * (8y * 8z ) = (8x * 8y) * 8z for all x, y, z in K.

(H2) (8x * 8y)V = 8yv * 8x v for all x, y in K.

(H3) 8x * 8e = 8e * 8x = 8x for all x in K.

(H4) e is in the support, supp(8x * 8yv), if and only if x = y.

(H5) supp(8x * 8y) is compact for all x, y in K.

(H6) The mapping (x, y) ---> supp( 8x * 8y) of K x K into the space of nonvoid compact subsets of K is continuous, where the latter space is given the "Michael" topology in [J], §2.5.

A locally compact group G is a hypergroup satisfying these axioms. In this case, 8x * 8y = 8xy for all x, y in G and XV is the inverse of x. Axioms (HI)-(H4) are clear. Note that axiom (H4) says that xy-l = e if and only if x = y. Axiom (H5) is very clear since each supp(8x * 8y )

consists of the single element xv. The technical axiom (H6) can also

Hypergroups and Signed Hypergroups 79

be verified in this setting. I will only give one dass of examples at the end of this section, but

many examples appear throughout this publication. A very key concept in locally compact groups and hypergroups is

that of an invariant measure. For the group case, Alfred Haar proved in 1933 that a (second countable) locally compact group has a leftinvariant measure m, which we now call a left Haar measure. Later authors proved that the left Haar measure is unique up to a positive constant. Left invariant me ans that m(xE) = m(E) for all x in C and all Borel sets E in C. This is equivalent to the requirement that 8x *m = m for all x in C, where the convolution here is extended in the natural way so as to apply even if m is an infinite measure. Similarly, a right Haar measure m' is one that satisfies m' * 8x = m' for all x in C. Of course, locally compact groups also have right-invariant Haar measures. We will return to this topic.

As in the group case, a measure m on a DJS-hypergroup K is called a left Haar measure if 8x * m = m for all x in K, with a similar definition for right Haar measure. Does every K have Haar measures? Each of the pioneers Dunkl, Jewett and Spector proved that every compact hypergroup K has a left Haar measure. I believe that each of them proved the same results for discrete hypergroups. In any case, the result is easy. The formula is m( {x}) = [8xv * 8x] ({ e })-l for all x in K.

The case for commutative hypergroups was substantially harder, but Spector [S] proved that every commutative hypergroup has a Haar measure. Incidentally, Spector uses weaker axioms than the DJSaxioms and pays aprice. He doesn't assurne axiom (H4). His proofs simplify somewhat if we use the DJS-axioms. Remarkably, the general quest ion of whether every DJS-hypergroup has a Haar measure is still open. This is certainly the biggest open quest ion in the subject. Note that, just as in the group case, when the Haar measures exist they are unique up to a constant. This was shown by Jewett P].

As in the group case, we say that a hypergroup is unimodular if the left and right Haar measures coincide. Groups that are either commutative, compact, or discrete are unimodular. Obviously commutative hypergroups are unimodular, and it is not hard to show that compact hypergroups are also unimodular. Surprisingly, on page 39 of Bloom and Heyer's book it is noted that "in contrast to the group case it is unknown whether all discrete hypergroups are necessarily unimodular." However, the recent paper [KW] contains a construction for a dass of non-unimodular discrete hypergroups! They arise as double-coset hypergroups induced by the transitive action of non-unimodular groups

80 K.A. Ross

of permutations on an infinite set. The most work has been done on commutative hypergroups for

the simple reason that they are easier to deal with. As in the case of groups, Fourier and Fourier-Stieltjes transforms playa big role. These are functions that, in the group case, are defined on the character group. In the case of a hypergroup K, they are defined on the space K" of all characters, which might or might not be a hypergroup in its own right. There are even three-element hypergroups K for which K" is not a hypergroup. This is an interesting issue and, for finite hypergroups, I will touch on it again in section 3.

A hypergroup character X on a hypergroup K is a bounded complex-valued continuous function that is not identically zero and satisfies

(i) X(XV ) is the conjugate of X(x) for all x in K, (ii) X(x * y) = X(x)X(y) for all x, y in K.

Each of these requirements deserves comment. With the other requirements, (i) holds automatically for locally compact groups because xx- 1 = e for all x. This remark is also true for compact hypergroups, but in general there are functions X that satisfy (ii) and not (i). Property (ii) needs clarification because x * y isn't defined. This very suggestive notation is due to Jewett. In general,

In the group case, this is f(xy) just like it should be. Many familiar results carry over to general and to commutative

hypergroups. I will only mention the Levitan-Parseval identity, wh ich I will state in the easy case that K is finite and commutative. Then K" has a Plancherel measure v, and

Parseval's identity. L If(x)1 2m(x) = L 1F"'(xWv(X). xEK XEK"

If m is normalized so that m(K) = 1, then v is normalized so that v(l) = l.

Let me end with a family of very simple hypergroups that will play a role in the next section. Here are all of the two-element hypergroups. For ° < () :S 1, the set K will be {O, I}, ° will serve as the identity and the involution will be the identity map. Since 0 is the identity, the three products 80 * 80 = 80 , 80 * 81 = 81 * 80 = 81 are automatic. The

only interesting product is

Since the coefficients have to be non negative and add to 1, we must have 0 ~ () ~ 1. Since 0 has to be in the support of 81 * 81 , we have to have () > O. It is easy to check that we get a hypergroup for each (). Note that for () = 1, we get the familiar two-element group Z(2).

Ze(2) has two characters, the identically 1 function and X where X(O) = 1 and X(l) = -(). The normalized Haar measure me is given by me(O) = ()/(1 + ()) and me(l) = 1/(1 + 0). Again, note that if () = 1, we get the correct characters and Haar measure on Z(2). If we think of () as varying continuously starting at 1, then we can view the family {Ze(2) : 0 < () ::; I} of hypergroups as adeformation of the group Z(2).

In the next section I will focus on the group Z(2)d of all d-tuples of O's and 1 'so This can be deformed into a big product of hypergroups, namely Ze(2)d. The hypergroup characters of Ze(2)d are easy to determine in terms of the characters of each factor Ze(2). And the Haar measure me on Ze(2)d isjust the product ofthe Haar measures on each factor. Thus

me(x) = Od-H(x) /(1 + O)d for all x in Ze(2)d.

Remember, x is astring of O's and l's; here H(x) is the number of these terms that are equal to 1.

2. Markov Chains and Deformations of Hypergroups

I am going to begin by discussing "random walks." The walk needs to be governed by so me rules: initially our walks will be in finite groups. Random, as the word suggests, means that at each step the walk will be governed by a probability Q. I will assurne that a particle starts at the identity of the group. Q will describe the various first steps and their probabilities. Q(n) will describe the position of the particle after n steps. It turns out that Q(n) is the convolution of Q n-times.

Let U be the uniform prob ability measure on a finite group G. This is, of course, Haar measure normalized to give the group total mass 1. It has been known since at least the 1940's that Q(n) converges to U unless there are obvious impediments, like the support of Q lies in a proper subgroup of G. In other words, if there were many particles, the particles would be evenly mixed after awhile. In the past fifteen years, under the guidance of the guru Persi Diaconis, there has been renewed

82 K.A. Ross

interest in studying how fast Q(n) converges to U. First, one needs a measurement of closeness. Here's a commonly accepted measurement, called the total variation distance:

IIQ(n)-UII = max{IQ(n)(A)-U(A)1 : A c G} = ~ L IQ(n)(Y)-U(y)l· yEG

Here are so me examples.

(a) Repeated shuffiing of a deck of cards can be viewed as a random walk on the large non-abelian group of all permutations of the deck. If one shuffies with the same randomness, then there is a unique probability Q that describes the shuffies .. Diaconis and his colleagues have studied the difficult quest ion of determining the rate at which IIQ(n) - UII converges to O. This depends, of course, on the choice of Q but there are interesting choices of Q and choices that reflect real shuffies. The papers [BD] and [AD] provide nice introductions to this subject. The book [D] is the basic "textbook" of the subject.

(b) Interesting random walks occur in much simpler abelian groups, like the cyclic group Z(q) on q elements. One natural Q is the "nearest neighbor random walk" where Q(l) = Q(q - 1) = 0.5. If q is even, then this lives on a proper subgroup. To avoid this sort of problem, we sometimes study the "nearest neighbor or stay at horne random walk." Here Q(l), Q(q - 1), and Q(O) may all be set equal to one-third.

Before going on, let me mention a powerful, but simple, tool that Diaconis and his co-workers have used. They call it the Upper Bound Lemma. They have aversion for non-abelian groups, but here I will only consider finite abelian groups. It is an easy consequence of Parseval's identity after bounding the norm by an ji2 norm.

Upper Bound Lemma. 41IQ(n) - UI1 2 :::; I:X#l IQA(X)1 2n.

The easiest non-trivial application is to Z(3) with the nearest neighbor random walk Q(l) = Q(2) = 0.5. We worked this out in [RX3] and found that IIQ(n) - UI1 2 :::; 2-2n- 1• For n = 10, this yields IIQ(lO) - UII :::; 0.000691. Direct calculation shows that IIQ(lO) - UII is approximately 0.000650.

(c) Now consider the cube Z(2)d of all d-tuples of O's and l's. For each x in Z(2)d, let H(x) be the number of coordinates of x with a 1. This is the Hamming distance from x to the origin. There's a natural graph of the group where one connects two elements if they differ in exact1y one coordinate, i.e., if H(x - y) = 1. A natural random walk is the "nearest neighbor random walk" where each of the d elements

with H(x) = 1 has prob ability l/d. The only problem is that there ean be parity problems. To avoid these we again use the "nearest neighbor or stay at horne" random walk. Le., Q(O) = l/(d + 1) and Q(x) = l/(d + 1) for all x with H(x) = 1. The Upper Bound Lemma ean be used to estimate IIQ(n) - UII. Some eombinatorial estimates are needed. This quantity goes to 0 exponentially with n. For example, if d = 2, we find that 41IQ(n) - UI12 ~ (1/3)2n-l.

We ean view every random walk as a Markov ehain. In fact, the transition matrix M eorresponding to Q is given by M (x, y) = 8x * Q (y) for all x, y in G. Here 8x denotes the point mass at x and * denotes eonvolution. Note that M(e, y) = Q(y), where e is the identity of the group. It follows easily by induetion that the matrix power satisfies Mn(x, y) = [8x * Q(n)](y) for x, y in G. If we want to study Markov ehains, there are many teehniques for studying them. However, if it turns out that the Markov ehain is aetually a random walk, then we also have the tools of elementary harmonie analysis at our disposal, as illustrated by the use of the Upper Bound Lemma.

Diaeonis and Hanlon [DH] studied some special Markov ehains called Metropolis Markov chains. As I will explain, the ones that they studied are based on familiar random walks on finite groups, but are not themselves random walks. Metropolis Markov ehains are so-named beeause of an old paper with four or five authors, among them Nicholas Metropolis and Edward Teller.

Metropolis Markov chains make sense on any finite set X. Let 7r be a probability on X with 7r(x) > 0 for all x in X. The Metropolis algorithm is a classical Markov ehain simulation method for sampling from 7r, which is effeetive when the ratios 7r(x)/7r(Y) are available. We begin with a "base chain" B(x, y) that is partly symmetrie, Le., B(x, y) = 0 if and only if B(y, x) = O. For us, B will be generated by a random walk. We will use the ratios

ry,x = [7r(y)B(y, x)]j[7r(x)B(x, y)],

where we deeree that this is 0 if B(x, y) = B(y, x) = O. Here is the Metropolis Markov chain:

M(x,y) = B(x,y)

= B(x, y)ry,x

= B(x, x) + E(x)

if y i- x and ry,x ~ 1;

if ry,x < 1;

if y = x,

where E(x) = L B(x, z)(l-rz,x), summed over all z f- x with rz,x < 1.

84 K.A. Ross

That is, E(x) is the value needed so that the sum of all M(x, y), over y in X, is 1.

The stationary distribution for this Markov chain will be the original prob ability 7r. Diaconis and Hanlon were interested in the convergence rates of these Markov chains, but note that now this may weIl depend on the starting point. So the variation distance is defined by

Consider again the nearest neighbor walk on Z(2)d : Q(x) = lid if and only if H(x) = 1. Note that the corresponding Markov chain B is given by

B(x, y) = Ox * Q(y) = Q(y - x) = lid if and only if H(x - y) = 1.

All we need now, to specify a Metropolis Markov chain, is the stationary distribution. Diaconis and Hanlon consider the measures 7ro defined on Z(2)d by

7ro(x) = (JH(x) 1(1 + (J)d, where 0< (J ~ 1.

When (J = 1, this is just the uniform prob ability measure on the group Z(2)d. Diaconis and Hanlon find the eigenvectors and eigenvalues of the Metropolis Markov chain M in terms of Krawtchouk polynomials and then establish the rate of convergence of IIMn(o,_) - 7roll. I won't give you the complicated results, but this will be small if n is roughly d log( d) + c, where c does not depend on d.

In [RX1], Daming Xu and I analyzed the Diaconis-Hanlon result from a little different point of view. First, recall that the hypergroups ZO(2)d have Haar measure given by mo(x) = (Jd-H(x) 1(1 + (J)d. This is very like the 7ro studied by Diaconis-Hanlon. Since H(x) counts the number of 1 's in x, and d - H (x) counts the number of O's in x, a simple change of variable, x --t 1 - x, in the Diaconis-Hanlon Markov chain changes their stationary distribution to the Haar measure. With this trivial change, we see that they estimated the rate of convergence of IIMn(l,_) - moll.

Since random walks on a finite group converge to the Haar measure of the group, unless there are obvious impediments, this suggests that perhaps M is, in fact, a random walk but on the hypergroup ZO(2)d. This is the case and the prob ability measure that generates this random walk is exactly the nearest neighbor randorn walk Q. The Metropolis Markov chains studied by Diaconis and Hanlon, which can be viewed


as deformations of the nearest neighbor random walk, are precisely the nearest neighbor random walks on the hypergroup deformations of the group Z(2)d.

The U pper Bound Lemma mentioned before carries over to hypergroups with no difficulty. It now reads

Upper Bound Lemma. 41IQ(n) - ml1 2 ::; L x#l IQ"(x)1 2n v (x).

Here v is the Plancherel measure on K". Since v is counting measure in the group case, this result is an exact generalization of the original U pp er Bound Lemma. For the nearest neighbor random walk Q on the hypergroup ZO(2)d, the summands in the Upper Bound Lemma are easily calculated. Some careful algebraic estimations then lead to exactly the same bounds that Diaconis and Hanlon obtained. We get one benefit, though, because it is now clear that the bounds work no matter what point our random walk starts at. That is, the bounds that Diaconis and Hanlon obtained for IIMn(o, _) -?Toll hold for IIMn(x, _)?Toll = IIMn(l - x,_) - moll·

Using the same methods, Daming Xu and I also obtained similar estimates for the Metropolis Markov chain associated with the nearest neighbor random walk on Z(3)d. In [RX2], we work hard to obtain similar results involving the groups Sn of all permutations of an n-element set. Again the basic random walk is the nearest neighbor random walk starting at the identity permutation. The nearest neighbors are transpositions. Diaconis and Hanlon studied the Metropolis Markov chain in this setting. They "lumped" the chain to the space K n of conjugacy classes. The Markov chains so obtained are still not random walks, but they are if we then deform K n into an object that isn't even a hypergroup. The objects are called signed hypergroups. They will be the topic in section 3. I won't return to the messy calculations necessary to establish rates of convergence, but I want to emphasize that our interest in signed hypergroups began here.

[RX3] contains a nice expository account of all of this, and much more. In particular, we work out the details for some Metropolis Markov chains on S~ and on si. Note that this article was published in the J. Math. Sciences in honor of my friend U. N. Singh.

3. Signed Hypergroups

As we all know, the theory of locally compact hypergroups is weIl established. The theory is quite rich and many generalizations from locally compact groups have been obtained. But even more general structures have been studied and found to be useful. Indeed, much

86 K.A. Ross

of the work of Berezansky and other former Soviet mathematicians were in a more general context. However, in this section I am going to focus on what are called signed hypergroups. My focus will be on axiomatics. The process is still ongoing in the sense that I still do not know the "right" axioms for a general locally compact (Hausdorff) signed hypergroup.

Recall that, in section 2, I indicated that my colleague Daming Xu and I were led to signed hypergroups because we were looking for ways to look at certain Markov chains as random walks. We were fortunate to leam about a very simple elegant system ofaxioms, due to Norman Wildberger [W], for what are now called finite signed hypergroups. This story is told in my Seattle conference survey [Ro], so I will only give enough here to make this report coherent. Let's look again at the key axioms for a DJS-hypergroup. We are given a locally compact Hausdorff space K with an involutive homeomorphism X -+ XV and a special element e. In addition

(H*) A continuous mapping (x,y) -+ 8x * 8y from K x K into M1(K), where M1(K) has the weak topology with respect to the space Cc(K) of continuous complex-valued functions with compact support. [convolution]

(H1) 8x * (8y * 8z ) = (8x * 8y ) * 8z for all X, y, z in K.

(H2) (bx * by)V = 8yv * bxv for all X, y in K.

(H3) 8x * 8e = be * 8x = bx for all X in K.

(H4) e is in the support Supp(bx * byv) if and only if X = y.

(H5) SUpp(bx * 8y) is compact for all x, y in K.

(H6) The mapping (x, y) -+ supp(8x * by) of K x K into the space of nonvoid compact subsets of K is continuous, where the latter space is given the "Michael" topology in [J], §2.5.

For a finite DJS-hypergroup, axioms (H5) and (H6) are not needed since they automatically hold. This leaves us with (H*) and (H1)-(H4), and the continuity condition in (H*) is no longer an issue. The only change needed for finite signed hypergroups is to weaken (H*) to

(SH*) For each (x,y) in K x K, 8x * 8y is areal (or signed) measure satisfying 8x * 8y (K) = 1,

and to add

(Pe) 8x v * 8x (e) > 0 for all x in K.

We need axiom (Pe) in order to show the existence of Haar measure. It's given by the same formula I mentioned for discrete hypergroups:


m({x}) = [8xv *8x ]({e})-1 for all x in K. I should mention right now that Margit Rösler weakens (SH) fur

ther and only requires that Ox * Oy be areal measure. However, she shows in her setting that she gets an invariant measure if and only if 8x * 8y (K) = 1. She weakens the hypothesis beeause in [R1] she studies a Laguerre eonvolution system and in [R2] she studies Bessel-type signed hypergroups that do not have this property. See [R3] and [R4] for more reeent papers by Rösler on signed hypergroups.

A merit of Wildberger's axioms is the following.

Duality theorem. If K is a finite commutative signed hypergroup, then K A is also a commutative signed hypergroup under pointwise operations and conjugation, and K is the character hypergroup of K A in a natural way. In particular, if K is a finite commutative hypergroup, then even though K A need not be a hypergroup, it is a signed hypergroup and its dual is K.

Moreover, one ean do "harmonie analysis" in the setting of these signed hypergroups. I would like to see the eorreet axioms for general loeally eompaet signed hypergroups. For one thing, they should be axioms that, when eaeh 8x * Oy is a prob ability measure, reduee to the axioms of a DJS-hypergroup. I will diseuss where we seem to stand.

Incidentally, every two-element signed hypergroup has the form Zo(2) = {O, I} where 81 * 01 = 000 + (1 - 0)81, where we now allow 0 to be any positive number.

I believe that I know the eorreet axioms for a diserete signed hypergroup. I believe that we need to reinstate (H5):

(H5) supp(8x * 8y) is compact [Le., finite] for all x, y in K,

and add

sup{[[8x * 8y [[ : x, y in K} is finite.

This is needed if one is going to get eonvolution inequalities that make L1 into a Banach algebra, LP into an L 1-Banach module, ete.; see §4 of [Ro].

Beyond the diserete ease, the situation is much less deaL In [Ro] I proposed some axioms for eompaet signed hypergroups. At the same time, Margit Rösler [R1] proposed a similar set ofaxioms for a-compact signed hypergroups. My student Adam Parr [P] is trying to find a good common set ofaxioms for arbitrary (locally eompact) signed hypergroups. The goal is to find axioms so that

(a) for discrete signed hypergroups, they agree with the axioms that

88 K.A. Ross

I've given;

(b) when each bx * by is a nonnegative measure, hence a prob ability measure, then the axioms agree with Jewett's axioms for DJShypergroups;

(c) for a-compact signed hypergroups, they are as compatible with Margit Rösler's axioms as possible when one adds axioms (SH*) and (H5) to her set ofaxioms.

I will set down Parr's tentative set ofaxioms. I will then compare them with Jewett's axioms and with Rösler's axioms. I will also discuss some of the issues involved. Here are Parr's axioms: We are given a locally compact Hausdorff space K with an involutive homeomorphism x -+ x v, a special element e, and an associative bilinear mapping (J-l, v) -+ J-l * v from M(K) x M(K) into M(K). In addition

(SH*) A continuous mapping (x, y) -+ bx *by from K x K to M(K), where M(K) has the weak topology Tb with respect to the space Cb(K) of all bounded continuous complex-valued functions on K.

(SHl) for each (x,y) in K x K, bx * by is areal (ar signed) measure satisfying bx * by (K) = l.

(SH2) (bx * by)V = byv * bxv far all x, y in K.

(SH3) bx * be = be * bx = bx for all x in K.

(SH4) z is in supp(bx * by ) if and only if y is in supp(bx v * bz ) for all x, y, z in K.

(SH5) supp(bx * by) is compact for all x, y in K.

(SH6) The mapping (x, y) -+ supp(bx * by) of K x K into the space of nonvoid compact subsets of K is continuous, where the latter space is given the "Michael" topology in [J], §2.5.

(SH7) sup{ Ilbx * by 11 : x, y in K} is finite.

(SH8) There exists a left invariant measure m on K with supp(m) = K and satisfying

(*) L J(x * y)g(y)dm(y)

= L J(y)g(XV * y)dm(y) for all x in K and J, gin Cc(K).

Remarks. (SH*) Jewett's version assurnes that the mapping (x, y) -+

bx * 8y from K x K to M+(K) is continuous, where M+(K) has the weak topology with respect to the space Cc(K) U {I}; he calls this the

"cone topology." This topology agrees with Tb on M+(K). This also agrees with the weak topology on Ml(K) with respect to Ge(K). Rösler assurnes that K is <T-compact and assurnes that the mapping (x, y) -+

8x * 8y from K x K to M(K) is continuous, where M(K) has the weak topology with respect to the space Go(K) of the bounded continuous complex-valued functions on K that vanish at infinity. Whatever the topology is on M(K), the goal is to have the convolution mapping (11" 1/) -+ J.L * 1/ continuous or separately continuous on some domain. Jewett gets it continuous in the co ne topology from M+(K) x M+(K) to M+(K). Rösler gets it separately continuous from M(K) x M(K) to M(K) by adding another axiom: given f in Ge(K) and x in K, each map y -+ fex * y) and y -+ f(y * x) is in Ge(K).

Parr gets the mapping (J.L,1/) -+ J.L * 1/ continuous from M(K) x M(K) to M(K). He used the strict topology on Gb(K) as a tool, and his work was motivated by help from Professor Ajit Iqbal Singh.

(SHl) As mentioned before, Rösler does not require 8x *8y (K) = 1 because she has important examples without this property. Of course, there isn't an invariant measure whenever this property fails.

(SH2) With the current set ofaxioms, this one is redundant. Rösler [R3] points out that (SH2) is a consequence of (*) in axiom (SH8).

(SH4) Note that, in the group case, this axiom merely states that z = xy if and only if y = x-l z. This axiom immediately implies axiom (H4), and the two axioms are equivalent for discrete signed hypergroups. Rösler does not assume either of these axioms. With her weaker axioms, in [R3] she is able to prove that if x is in X and e does not belong to any suppe 8x * 8y) for y =1= x v, then e does belong to supp(8x * 8xv).

(SH5) This useful axiom was made by Jewett, but not by Spector. Since it does not hold in an interesting examples, Rösler does not assume this axiom either.

(SH6) Rösler avoids this axiom. Her substitute seems to be the requirement that for each compact subset F of K and each f in Ge(K), the union of an of the supports of an of the translates y -+ f (x * y) and y -+ f(y * x), as x ranges over F, is precompact.

(SH8) This is a distressing axiom because it is not an axiom for DJS-hypergroups. Moreover, Jewett [J], 5.lD, proves (*) holds provided that the hypergroup has a Haar measure. Of course, many results require the extra assumption that Haar measure exists. While discrete signed hypergroups have a Haar measure using the extra axiom (Pe),

90 K.A. Ross

without axiom (SH8) it is not known whether compact or commutative hypergroups must have a Haar measure. Parr has examined analogues to axiom (Pe) to see if they might imply the existence of Haar measure. In [Ro] , where I set down some tentative axioms for compact signed hypergroups, I avoided (*) by giving an awkward replacement. It has the advantage, though, that I was able to verify it more easily in some examples. The property was created to make the proof of 5.1D in [J] carry over to my setting.

References

[AD] D. Aldous and P. Diaconis, Shuffling eards and stopping times, Amer. Math. Monthly 93 (1986), 333-348.

[BD] D. Bayer and P. Diaconis, Trailing the dovetail shuffle to its lair, Annals of Applied Prob. 2 (1992), 294-313.

[BH] W. R. Bloom and H. Heyer, Harmonie Analysis of Probability Measures on Hypergroups, de Gruyter Studies in Mathematics, 1995.

[BK1] Yu. M. Berezansky and A. A. Kalyushnyi, Hypercomplex systems and hypergroups: connections and distinctions, Contemporary Math. AMS, 1995 (Applications of Hypergroups and Related Measure Algebras-Seattle conference, July 31-August 6, 1993), 21-44.

[BK2] Yu. M. Berezansky and Yu. G. Kondratiev, Biorthogonal systems in hypergroups: an extension of non-Gaussian analysis, University of Bielefeld preprint.

[CGS] Applications of Hypergroups and Related Measure Algebras (Seattle conference, July 31-August 6, 1993), Contemporary Math. AMS, 1995. See also [BK1], [K], [R1] and [Ro].

[D] P. Diaconis, Group Representations in Probability and Statistics, Institute of Mathematical Statistics (1988), Hayward, California.

[DH] P. Diaconis and P. Hanlon, Eigen analysis for some examples of the Metropolis algorithm, Contemporary Math. 138 (1992), 99-117.

[H] Probability Measures on Groups and Related Structures, Eleventh Proceedings- Oberwolfach 1994, edited by H. Heyer, World Scientific Publishing Co. See also [KW] and [R2].

[J] R. 1. Jewett, Spaces with an abstract convolution of measures, Advances in Math. 18 (1975), 1-101.

[K] T. H. Koornwinder, Compact quantum Gelfand pairs, Contemporary Math. AMS, 1995 (Applications of Hypergroups and


Related Measure Algebras-Seattle conference, July 31-August 6, 1993), 213-235.

[KW] V. A. Kaimanovich and W. Woess, Construction of discrete, non-unimodular hypergroups, Proc. Oberwolfach 1994 (Probability Measures on Groups and Related Structures), 196-209.

[L] R. Lasser, Orthogonal polynomials and hypergroups, Rend. Mat. 3 (1983), 185-209.

[P] A. Parr, Signed hypergroups, thesis, University of Oregon, 1997.

[R1] M. Rösler, Convolution algebras which are not necessarily positivity-preserving, Contemporary Math. AMS, 1995 (Application of Hypergroups and Related Measure Algebras-Seattle conference, July 31-August 6, 1993),299-318.

[R2] M. Rösler, Bessel-type signed hypergroups on R, Proc. Oberwolfach 1994 (Probability Measures on Groups and Related Structures), 292-304.

[R3] M. Rösler, On the dual of a commutative signed hypergroup, Manuser. Math. 88 (1995), 147-163.

[R4] M. Rösler, Partial characters and signed quotient hypergroups, preprint.

[Ro] K. A. Ross, Signed hypergroups-a survey, Contemporary Math. AMS, 1995 (Applications of Hypergroups and Related Measure Algebras-Seattle conference, July 31-August 6, 1993), 319-329.

[RX1] K. A. Ross and D. Xu, Hypergroup deformations and Markov chains, J. Theoretical Probability 7 (1994), 813-830.

[RX2] K. A. Ross and D. Xu, Metropolis Markov chains on Sn and signed hypergroup deformations. Unpublished (1993).

[RX3] K. A. Ross and D. Xu, Some Metropolis Markov chains are random walks on hypergroups, J. Math. Seien ces 28 (1994), 194-234.

[S] R. Spector, Mesures invariantes sur les hypergroups, Trans. Amer. Math. Soc. 239 (1978), 147-165.

[W] N. J. Wildberger, Duality and entropy for finite abelian hypergroups, preprint, University of New South Wales.

Email: [email protected]; Fax: (541) 346-0987; Department of Mathematics, University of Oregon, Eugene, OR 97403-1222

Three lectures on Hypergroups

Delhi, December 1995

Alan L. Schwant'

Abstract

LECTURE 1. Cosines, Legendre polynomials, and Bessel functions are examples of eigenfunctions of Sturm-Liouville problems which are also characters of hypergroups. There are many additional examples from the classical special functions. Indeed, it is possible to give conditions on a Sturm-Liouville problem so that its eigenfunctions must be characters of a hypergroup. The converse will also be discussed. If a hypergroup consists of measures on areal (compact or not) interval, then with adequate regularity conditions, it must be the case that the characters of the hypergroup are eigenfunctions of a Sturm-Liouville problem.

LECTURE 2. A family of orthogonal polynomials on an interval I may be the characters of a hypergroup of measures supported on I (which would be called a continuous polynomial hypergroup), and the family mayaiso supply the characters of a hypergroup of measures on the discrete set {O, 1, 2, ... } (which would be called a discrete polynomial hypergroup). The entire category of continuous polynomial hypergroups can be explicitly described, but the full category of discrete polynomial hypergroups has not yet been characterized, though there are some fairly general theorems.

LECTURE 3. The same issues in the second talk raise analagous questions for multivariate orthogonal polynomials. A family of multivariate orthogonal polynomials is a much more subtle object than a family of one-variable orthogonal polynomials. Some progress has been made in the classification problem, and these results will be discussed as well as recently discovered examples.

*The preparation of these lectures took place during the tenure of NSF grant DMS-9404316

94 A. Schwartz

Introd uction

I have prepared these informal not es to accompany my lectures at the International Conference on Harmonie Analysis, Delhi, December 1995. The purpose of the lectures is to illustrate the interactions among hypergroups, polynomials, and differential equations. The notes are intended as an expanded version of the lectures; neither the not es nor the lectures are offered as a complete survey of the subject, but rat her in the hope of whetting the appetite and providing some useful entry points into the literature. I would like to take this opportunity to apologize to the many authors whose work would be included in a more complete survey, but are not included here for lack of space. Some proofs of typical results will be outlined, but the careful reader will have to consult the original memoirs for complete proofs.

The three sections of the not es correspond roughly to the three lectures and provide somewhat more detail than can be covered in an hour.

I have taken the liberty of including many more bibliographie items than are cited in the exposition below in the hope that the reader may find this a convenient list of references to explore the subject.

We mention here three references that are of particular interest to anybody seeking to learn more about this field.

• Dunkl's article [Dun73] is a relatively short exposition in a system somewhat more general than a commutative hypergroup, but it is a nice introduction to the ideas and methods of the subject.

• Jewett's article [Jew75] is a textbook that thoroughly covers the basic properties of hypergroups (although Jewett uses the term "convo" for "hypergroup").

• Litvinov's survey [Lit87], is especially valuable for its broad coverage of much of the work done up to 1985, including such related topics as hypercomplex systems.

• The book of Bloom and Heyer [BH95] gives a unified treatment of the subject including a large body of work on prob ability in hypergroups; the book includes many examples and a wealth of references.

• The proceedings of a 1993 conference "Applications of Hypergroups and Related Measure Algebras," is published as [CGS95].

Three Lectures on Hypergroups 95

Finally, we mention that a number of research quest ions are gathered in aseparate section near the end of these notes.

1. Differential Equations and Hypergroups

As usual, C(H) denotes the continuous functions on H, M(H) the bounded Borel measures on H, and M1(H) the prob ability measures on H. The support of a measure /-l is denoted supp(/-l), and the unit point mass at x will be denoted t5x . H will often be one of the following: 1= [-1,1], J = [0,1f], lR+ = [0,(0), No = {O, 1,2, ... }.

1.1. Legendre polynomials and beyond

We begin with an old example to illustrate how a family of functions, in this case orthogonal polynomials, can give rise to a measure algebra. I.I. Hirschman, Jr., pointed out in 1956 [Hir56b, Hir56a] that the structure for harmonic analysis exists in a setting where certain orthogonal polynomials could play the role of the exponentials in classical Fourier analysis. For example, consider the Legendre polynomials {Pn}nENo' These are orthogonal with respect to Lebesgue measure on I and are normalized by requiring Pn (l) = 1. The Legendre polynomials satisfy a product formula:

(-l<x,y<l)

with

the first value being taken if and only if 1 - x2 - y2 - z2 + 2xyz > 0. Obviously K(x, y, z) ~ ° and since Po(x) = 1 it follows from the product formula that 1 K(x, y, z)dz = 1.

For J, 9 E P = L1(I, dx) define

U * g)(z) = 11 K(x, y, z)J(x)g(y) dx dy

so that

1U * g)(x)Pn(x) dx = [1 J(x)Pn(x) dX] . [1 g(x)Pn(x) dX] ,

96 A. Schwartz

and it follows that (LI, *) is a Banach algebra. The operation is easily extended to the point masses by defining

d(ox * Oy)(z) = K(x, y, z) dz,

and

(-l<x,y<l),

(x E I).

Finally, if J-L, /J E M(I), define J-L * /J by its action on an arbitrary continuous function:

(J E C(H)).

Actually, Hirschman discusses these constructions not just for the Legendre polynomials, but for the ultraspherical polynomials {pJol(x)} that are orthogonal on 1 with respect to the measure (1- X2)0-~ dx; the Legendre polynomials are ultraspherical polynomials with 0: = 1/2. The polynomials R~ol(x) = pJol(x)/pJol(l) are used in place ofthe Legendre polynomial Pn (x). In that case

{ 21-20(1 _ x2 _ y2 _ z2 + 2xYZ)0-1

K(ol(x, y, z) = ~2(0:)[(1 - x2)(1 - y2)(1 - Z2)]0-~ ,

with the first value being taken if and only if 1- x2 - y2 - Z2 + 2xyz > O. So for each 0: 2: -1/2 Hirschman obtains a measure algebra that we

denote (1, *0)' It is important to note *0 is a distinct convolution for each 0: 2: -1/2, hence a continuum of Banach algebras is built on the single Banach space M(I). The algebraic structure does not depend on any arithmetic in the underlying space 1.

In the special case when n = 20: + 2 is an integer, this structure can also be inherited from a group, since (1, *0) is isometrieally isomorphie to the subalgebra of M(SO(n)) consisting ofmeasures that are bi-invariant with respect to the action of SO(n -1), or equivalently the measures on the unit sphere in jRn that are invariant under all rotations that leave some designated point fixed. So (1, *0) interpolate these in some sense.

For most values of 0: the group structure is absent, yet there is still an adequate structure in (1, *0) to define and study objects like Fourier multipliers, maximal functions, and a Littlewood-Paley theory (see [CS77] and the references cited there).

1.2. Hypergroups

(1, *o,) is an example of a hypergroup. Roughly speaking, a hypergroup is a measure algebra that has many of the useful properties associated with the convolution measure algebra of a group, but no presumed algebraic structure for the underlying space. For the sake of clarity we will first give the hypergroup axioms as they apply to the case where the underlying space H is compact; this will be followed by the modifications for the locally compact case. Thus we assurne M(H) is a Banach algebra with product *. Then (H, *) is a hypergroup (or hypergroup measure algebra) if it has the following properties:

(H1) If f-l and v are prob ability measures on H, then so is f-l * v.

(H2) There is an element e E H such that 8e *f-l = f-l*8e for all f-l E M(H).

(H3) There is a continuous involution x t--+ XV (XVV = x) such that e E supp8x * 8y if and only if y = xv.

(H4) (f-l * v)V = V V * f-lv where f-lv is defined by JH f(x) df-lV(x) = JH f(xV) df-l(x).

(H5) (x, y) t--+ supp(8x * 8y) is continuous with an appropriate topology for the space C(H) of compact subsets of H.

(H6) (f-l, v) t--+ f-l * v is weak-* continuous.

The "appropriate topology" in (H5) is the Michael topology [Jew75, §2.5] which has a sub-basis consisting of the sets

Cu(V) = {K E C(H) : K n U -I- 0 and K c V}

where U and V are arbitrary open subsets of H. If H has a metric p, this topology is equivalent to the topology given by the Hausdorff metric on C(H), which is defined for A, B E C(H) by

p(A, B) = inf{r : Ac Vr(B) and Be Vr(A)}

where Vr(E) = {y EH: p(x, y) < r for so me x E E}. A proof of the equivalence is contained in [KS95, Lemma 4.1].

In the locally compact case, we must add to (H5) the requirement that supp(8x * 8y ) is compact, and in (H6) "weak-* continuous" must be replaced by "positive continuity," which requires that for each non-negative compactly supported f E C(H), the mapping (f-l, v) t--+ JH f d(f-l * v) is continuous when restricted to positive measures in M(H).

When H is discrete, for instance H = No, (H6) can obviously be dropped and (H5) can be replaced by

98 A. Schwanz

(H5) supp(ox * Oy) is finite.

The definitive set ofaxioms was given first by Jewett in his encylopaedic article [Jew75]. Jewett calls these objects "convos" and, ironically, never once uses the term "hypergroup."

We also need the following definitions: m is a Haar measure for the hypergroup (H, *) if for every x E H, m * Ox = Ox * m = m; m is a left Haar measure if at least the second equality holds. cjJ E C(H) is a character of (H, *) if cjJ is bounded, cjJ( x V) = cjJ( x) and

(x, y EH).

(H, *) is a Hermitian hypergroup if XV = x for every x E H. If (H, *) and (K, *) are hypergroups, we say they are equivalent or identical up to a change of variables if there is a homeomorphism A from H onto K such that if x = A-1(s), and y = A-1(t),

LI d(os * Ot) = /)1 0 A) d(ox * Oy) (f E C(K)).

When this happens we write (K,*) = A(H,*). In this case we have SV = A(xV ), and cjJ is a character of (H, *) if and only if cjJ 0 A-1 is a character of (K, *). If Hand Kare subsets of vector spaces and A is an affine mapping (i.e., x !---; A(x) - A(O) is linear), we say Hand K are linearly equivalent. Thus, for instance, any hypergroup (H, *) with H C ]R2 is linearly equivalent to a hypergroup with identity element (0,0).

The measure algebra of a group with identity e is an obvious example of a hypergroup; convolution is defined by Ox * Oy = Oxy, and involution by XV = x-I, the group inverse of x. In the example, in §1.1 above, (1, *a) is a Hermitian hypergroup with e = 1 and XV = x. The characters are {R~a)(x)} and the orthogonality measure (1-x2)a-~ dx is Haar measure. Jewett [Jew75] obtains most of the basic properties necessary to do analysis in hypergroups. In an earlier article Dunkl [Dun73] had shown that a hypergroup (H, *) has a Haar measure if H is compact and * is commutative. Jewett [Jew75] shows that (H, *) has a Haar measure when H is compact, and that it has a left Haar measure when H is discrete. Spector [Spe78] showed (H, *) has a Haar measure when * is commutative. Both Dunkl and Spector use definitions for hypergroup that are slightly different from Jewett's, but their results still apply to the object of Jewett's convos. This is one of the reasons that the object we call a "hypergroup" is now often referred to as a "DJS hypergroup."

1.3. Spherically symmetrie random walks

Another early example of a hypergroup arose in probability theory. This example is the subject of a 1963 article by J. F. C. Kingman [Kin63]. Consider a pair of independent random variables X and Y in lR2 , with lengths X and Y, but with direetion uniformly distributed. The sum Z = X+ Y also has uniformly distributed direetion, but its length Z = IZI is a random number in the range IX - YI ::; Z ::; X + Y. In general, if X and Y are independent random variables in lR2 with uniformly distributed direetion, but with lengths X and Y having prob ability distributions J-l, v E M1(lR+), then Z is a random variable in lR+ with a probability distribution depending on J-l and v, denoted by J-l ° v, and we write Z = X EB Y. The operation ° is readily extended to all of M (lR+) so that (M (lR+), 0) beeomes a hypergroup measure algebra that is isometrieally isomorphie to the subalgebra of the group eonvolution algebra (M(lR2), *) eonsisting of the measures invariant with respeet to rotations of the plane. The eharaeters are indexed by lR+ and given by epy(x) = Jo(xy) where Jo is the Bessel function of the first kind of order 0:. These satisfy a product formula that yields

so that the useful substitute for the eharaeteristie function of the random variable X is x(y) = JR+ epy dJ-l. The product formula for the Bessel functions also ensures the fundamental property of eharacteristie equations XEl7Y = xy when X and Y are independent random variables in lR+.

Kingman aetually deseribes a eontinuum of Hermitian hypergroups (lR+, °0 ) (of course, he never uses the word "hypergroup"). The identity element is 0, and the eharaeters are given by epy(x) = Jo(Yx) = 2°f(0: + 1)(yx)-oJo(Yx) for y E lR+.

When n = 20:+2 is an integer, (lR+, °0 ) is isometrieally isomorphie to the subalgebras of rotation invariant measures on lRn . There is again no useful algebraie strueture in the underlying spaees. Nevertheless Kingman is able to define random walk and Brownian motion, and obtain a law of large numbers, a eentrallimit theorem, a reeurrenee theorem, and eharaeterizations of infinitely divisible and stable distributions. When n = 20: + 2 is an integer, all of this is an inheritanee from the group strueture on lRn , but Kingman obtains his results for all real 0: ~ -1/2 with no referenee to the group ease exeept for inspiration.

100 A. Schwartz

1.4. Differential equations

The examples discussed above have the property in common that the characters are eigenfunctions of a second-order linear differential operator. In the case of the ultraspherical example, it is convenient to make the change of variables x = cos Band write u~<» (B) = R~<» (cos B). These functions (for n E No) constitute a complete set of eigenfunctions for a Sturm-Liouville problem on [0, n]:

y'(O) = y'(n) = 0 , (1.1)

with p( B) = (sin B)<>. The modified Bessel functions J<> (yx) are the eigenfunctions for the following Sturm-Liouville problem on lR+:

(Ay')' + AAy = 0 y'(O) = 0 (1.2)

with A(x) = x2<>+l. This leads us to ask about the following:

Sufficient conditions. Which Sturm-Liouville problems have eigenfunctions that are characters of a hypergroup? In such cases, the full structure of hypergroups is available to elucidate expansions in terms of the eigenfunctions.

Necessary conditions. Which hypergroups on real intervals have characters that are eigenfunctions of Sturm-Liouville problems? In such cases, the differential equation yields detailed information about the characters.

1.5. Sufficient conditions The earliest relevant result we know of is Chebli [Che72]. He considers

the differential equation (1.2) with A satisfying the following conditions:

1. A(O) = O.

2. A(x) > 0 for x > O.

3. A'(x)jA(x) = a:x-1 + B(x) where B is continuous at O.

4. A(x) increases to 00 as x increases to 00.

This leads to the following result:

Theorem 1.1. Let a(x, y) = A(x)A(y). Suppose f is positive and twice continuously differentiable on lR+ and u(x, y) satisfies

(aux)x = (auy)y,

Then u(x, y) 2: O.

u(x,O) = f(x), uy(x,O) = O.

This gives rise to a hypergroup on lR+ with characters that satisfy (1.2) by a route similar to that which will be outlined below for the compact case.

Since 1972 a variety of similar results have emerged as weIl as many in the compact case; several of the latter are discussed and compared in [CS90b]. We outline the path given in [CS90b], which leads from a general Sturm-Liouville problem (1.1) to the corresponding hypergroup. We begin with some assumptions on p.

1. pis positive and continuous on (0,7r).

2. p(7r - s) = p(s).

3. p'(s)j p(s) is non-increasing in (0,7r).

4. p( s) = (sin s)' g( s) for some 'Y ~ 0. 9 must be real-analytic at ° and have p continuous derivatives on (0,7r) where p ~ max(ry + ~, 2).

It follows that the eigenvalues of (1.1) form an increasing sequence ° = /-Lo < /-LI < /-L2 < .... Let Yk be the eigenfunction corresponding to /-Lk normalized to Yk(O) = 1 so that, in particular, rlYo(s) = 1. Then

(1.3)

and {Yd is a complete orthogonal family for L2([0, 7r], p2(S) ds). The key fact is that {Yd satisfies a product formula:

Theorem 1.2. Por each s, t E J = [0,7r] there is f7s ,t E MI(J) such that

(k E No).

2. SUPPf7s ,t C [Is - tl, 7r -Is + t - 7r1]·

3. l(s, t) = fJ 1 df7s ,t is continuous on J x J.

Thus there is a Hermitian hypergroup (J, *) with character set {Yk}, Haar measure p2(S) ds, e = ° and convolution 01 two meaures /-L, v E M(J) defined by their action on 1 E C( J):

In particular, 8s * 8t = f7s ,t.

The proof of the theorem is based on two lemmas. Let P be the trigonometrie polynomials, that is, the spaee of finite linear eombinations from {Yk}'

102 A. Schwanz

Lemma 1.3. P is dense in C( J) in the topology of uniform convergence.

Proof. The differential equation can be used to show that if f has 2p continuous derivatives and compact support in (0,7r), then the Fourier series of f with respect to {Yd converges absolutely and uniformly to f .

• Lemma 1.4. If

n

f(s) = L CkYk(S) ~ 0, (s E J) (1.4) k=Q

then n

f(s, t) = L CkYk(S)Yk(t) ~ 0, ( ( s, t) E J x J). (1.5) k=Q

Proof. For 0 ~ a ~ b ~ 7r /2 let fJ.( a, b) be the tri angle with vertices (a, b), (a-b, 0), and (a+b, 0). We begin by assuming that (1.4) holds with > in place of ~; we will show that (1.5) holds in fJ.( 7r /2, 7r /2). Assume by way of contradiction that (1.5) fails at some point of fJ.(7r/2,7r/2). Then it is possible to select P = (~,'TI) E fJ.(7r/2,7r/2) such that f(~,'TI) = 0 but f(s, t) > 0 if (s, t) E fJ.(~, 'TI) - {P}.

The rest of the argument is buHt around a related hyperbolic Cauchy problem (as in Chebli's theorem). Let W(s, t) = p2(s)p2(t). f(s, t) satisfies

f(s,O) = f(s), ft(s,O) = O.

Now let C = (~ - 'TI,O) and D = (~+ 'TI,O) then Green's Theorem yields

0= rr [(Wfs)s - (Wft)t] dsdt J J tl(~,l1) = r (W fs dt + W ft dt) = - [r + r ] W df. ktl~~ kp kp

Integration by parts yields

2W(P)f(P) = W(C)f(C) + W(D)f(D)

+ r f(Wt + Ws) dt + r f(Wt - Ws) dt , JcP JDP

which implies f(P) > O. This contradicts the original assumption.

Now ifwe simply assume (1.4) and let f be any positive number, then

f, = f + f = f + fYo > 0,

thus for any (8,t) E Ö(7r/2,7r/2),

f,(8, t) = f(8, t) + fYo(8)YO(t) = f(8, t) + f > 0,

whence f(8, t) 2: 0 for (8, t) E Ö(7r/2,7r/2). Equation (1.5) is now obtained on all of J x J by employing the symmetries f(t, 8) = f(8, t) and f(7r - 8, 7r - t) = f(8, t) (recall (1.3)). •

Proof of theorem. If f is a non-negative function in C(J), then o ::; f(8) ::; Ilflloo, so Lemma 1.4 shows 0 ::; U(8, t) ::; Ilflloo. Thus (J s,t E MI (J). The other properties are all readily verified. •

Other results can be obtained by exploiting the maximum principal for hyperbolic equations or using a method devised by C. Markett [Mar89, CMS91, CMS92, CMS93] based on the Riemann integration method for such equations. This latter method has the advantage of yielding an explicit description of (Js,t and showing that it is absolutely continuous for 0< 8, t< 7r. That method yields product formulas (and the associated hypergroups) on an interval with modified Bessel functions [Mar89] and spheroidal wave functions [CMS93] as characters.

1.6. N ecessary conditions

The quest ion here is when a one-dimensional hypergroup (H, * ) (meaning one in which H is homeomorphic to the circle or areal interval) has characters tjat are eigenfunctions of a Sturm-Liouville problem. The first piece of the problem is to obtain some information ab out onedimensional hypergroups. All such hypergroups are commutative, and there are essentially four types [Sch88, Zeu89]:

1. H = lR and * is classical convolution. In this case e XV = -x.

o and

2. H = 1I' (where 1I' is the unit circle in the complex plane) and * is the classical convolution. In this case e = 1 and ZV = z.

3. His a compact interval, eis one endpoint, and XV = x.

4. H is a half-open interval (not necessarily bounded), e is the endpoint, and XV = x.

104 A. Schwartz

1. 7. Jacobi type hypergroups

We will concentrate on the situation where H is a compact interval; indeed, it is no loss of generality to require H = J = [0,7r]. We will give sufficient conditions on a hypergroup (J, *) so that its characters are exactly the eigenfunctions of a Sturm-Liouville problem (1.1). With each such hypergroup we will associate a pair of parameters (a, ß) that can be used to help describe the properties of the hypergroup. The definition is somewhat complicated, but it indudes (up to a change of variables) every hypergroup (H, * ), of which we are aware, where H is a compact interval. The discussion below is outlined from [CS90aJ.

We begin by requiring that (J, *) satisfies the following differentiablity condition: If f has p continuous derivatives and is compactly supported in (0,7r), then

has bounded pth-order derivatives on the interior of J x J. For f-t = 1, 2 let kl-' be the largest positive integer such that

When kl-' is finite,

MI-'(s, t) = AI-'(t)Sk" + o(Sk,,).

(J, *) is a Jacobi type (a, ß) hypergroup if

1. k1 = k2 = 2.

2. (sint)A1(t) is differentiable on J.

3. A2(t) is a positive constant a2.

4. a and ß satisfy

sin t 1 lim -A1(t) = a + -2

t-+O+ a2 and

sin t 1 lim -A1(t) = -(ß + -2) .

t-+7r- a2

The conditions are not strict as they seem, since a much larger dass of hypergroups are equivalent to Jacobi type hypergroups by a change of variable, so that it is really only necessary that k1 be finite, that (sint)A1(t) and A2(t) can be extended to be positive differentiable functions on J, and that A2 (t) be positive.

The definition is motivated by the example of the Jacobi polynomials p~a,ß), which are orthogonal on I with respect to the weight (1 - x)a (1 + x)ß. These include the ultraspherical polynomials since

R~a) = R~a-~,a-~). For many values of (0:, ß) the Jacobi polynomials normalized to the value 1 at x = 1 are the characters of a hypergroup (which will be discussed in the next lecture). That hypergroup with the change of variables x = cosB satisfies (1)-(5). The functions <Pn(t) = p~a,ß)(cost)/p~a,ß)(l) are the eigenfunctions of the Sturm-Liouville problem (1.1) for p = (sint/2)a+~(cost/2)ß+~.

Now assurne (J, *) is a Jacobi type (0:, ß) hypergroup. Let

p(t) = cexp -- dr , (l t Al (r) ) 7r /2 a2

where c is chosen so that J; p2(t) dt = 1. The eigenvalues of (1.1) can be arranged in an increasing sequence Ao < Al < A2 < ... ; let <Pk be the eigenfunction corresponding to Ak and satisfying <Pk(O) = 1; these are the characters of (J, *). We assurne 0:, ß ;::: -1/2. Then it is possible to obtain some precise estimates on the characters. First we introduce several constants. There are positive constants aa and aß such that

lim (sint)-ß-~p(t) = aß. t~7r-

Let Ea = 2ar(0: + l)aa, Eß = 2ßr(ß + l)aß' and E = Ea/ Eß. Then it is possible to obtain the following information about (J, * ):

1. p2 (t) dt is Haar measure for (J, * ).

2. 0:;::: ß ;::: -1/2.

3. limk-->CXl( _l)k ka- ß <Pk( 7r) = E.

6. There is K > 0 such that <Pk(t) ;::: ~ provided kt ::::; K.

7. There are constants C and CE for any E > 0 such that

106

where

{Ck- I

/h(s)/ = Cf lnk 1

k(sk)O:+"2

There is a similar estimate at 1L

A. Schwartz

0::; s ::; l/k

k- I ::; s ::; 7r - E.

These properties facilitate a study of the harmonie analysis of (J, *), so that one can obtain, for instance, a useful analogue of the HardyLittlewood maximal inequality [CS89J.

2. Orthogonal Polynomials and Hypergroups

2.1. Ultraspherical polynomials, again

Hirschman [Hir56b, Hir56a] also discusses an operation on fl = M(No) based on the linearization formula for ultraspherical polynomials:

n+m R~O:)(x)R~)(x) = L c~,mR~O:)(x). (2.1)

k=ln-ml

The explicit formula quoted by Hirschman shows c~ m 2: 0, and if we set x = 1 in (2.1) we obtain LkENo c~,m = 1. Thus if a,' b E fl = M(No) we can define their convolution a *0: b by

00 00

(a *0: b)k = L L c~,manbm n=Om=O

or equivalently for each x EI,

It is easy to check that M(No) with this operation is a hypergroup which we denote by (No, *0:)' Thus Hirschman describes two hypergroups (1, *0:) and (No, *0:) that have many properties analogous to the classical convolution algebras of measures on the circle group and its dual on the group of integers.

It is important to note *0: is a distinct convolution for each 0: 2: -1/2, hence a continuum of Banach algebras is again built on the single Banach space M(No). The algebraic structure does not depend on any arithmetic in the underlying space No.


2.2. Orthogonal polynomials

It is possible to formulate linearization and product formulas for general families of orthogonal polynomials and perhaps to imitate the construction of the two measure algebras as in the ultraspherical case. But there can be no guarantee in general that these operations are truly well-defined. We are interested in when the constructions actually yield hypergroups.

Thus suppose m E M1(R); let H = suppm. {Pn } is a family of orthogonal polynomials with respect to m if for each n, Pn is a polynomial of degree n that satisfies

(m=0,1, ... ,n-1).

These conditions determine Pn up to a non-zero constant. If xo is chosen so that none of the polynomials vanishes at Xo, we can define the normalized polynomials

so that

A family of orthogonal polynomials always has a linearization formula

where

n+m RnRm = L C~,mRk'

k=O

1HRkRnRm dm 11 R~dm

Clearly, the orthonormality of {Rn} implies C~ m = 0 if k > n + m or n > m + k or m > n + k; that is, the linea~ization formula can be rewritten

n+m RnRm = L C~,mRk' (2.2)

k=ln-ml

Evaluation of (2.2) at x = Xo yields

n+m

L c~,m = l. (2.3) k=ln-ml

If in fact

c~,m 2': 0 , (2.4)

108 A. Schwanz

it is possible to define the product of two sequences a, bE M(No) = fl by setting

00 00

(a*b)k = L L c~,manbm. n=Om=O

This is equivalent to saying

at least for finitely supported sequences a and b. Equations (2.3) and (2.4) guarantee that M(No) is a Banach algebra. In fact, (No,*) is a Hermitian hypergroup with identity element e = 0 and each character has the form

n=O provided z E C satisfies lub IRn(z)1 < 00. Such a hypergroup is called the discrete polynomial hypergroup associated with {Rn}.

A family of orthogonal polynomials always has a linearization formula, but there is no guarantee that it will have a product formula, which is the continuous analog of the linearization formula. That is, we say {Rn} has a product formula if for each pair x, y E H, there is a fixed measure O'x,y E M(H) such that for every n E No

(2.5)

If 1I00x,yll are uniformly bounded, it is possible to define a continuous product * on M (H) by the formula

L fd(J.l * l/) = L L L f dO'x,y dJ.l(x) dl/(Y) ,

so that, in particular, 8x * 8y = O'x,y. If O'x,y is a non-negative measure, evaluation of (2.5) for n = 0 shows that O'x,y E M1(H) so that M(H) becomes a Banach algebra with operation *. It also follows from (2.5) evaluated at x = Xo that O'XO,y = 8y, so that if in fact this Banach algebra is a hypergroup, it is Hermitian with identity element e = Xo. In that case we say that {Rn} has a hypergroup product formula and we call (H, *) the continuous polynomial hypergroup associated with {Rn}.

When a family of orthogonal polynomials has both a continuous and a discrete hypergroup associated with it, the two hypergroups are in some sense dual to one another in analogy with the way that the circle and the integers are a pair of dual groups.

We ask the following questions:

(Q1) Which hypergroups (No,*) are in fact discrete polynomial hypergroups?

(Q2) Which families of orthogonal polynomials give rise to discrete polynomial hypergroups?

and the corresponding quest ions on the continuous side:

(Q3) Which hypergroups (H, *) with H E ffi. are in fact continuous polynomial hypergroups?

(Q4) Which families of orthogonal polynomials give rise to continuous polynomial hypergroups?

2.3. Jacobi polynomials The normalized Jacobi polynomials R~a,ß) have product and lineariza

tion formulas for many values of the parameters. The region in the (a, ß)plane where this happens was identified by Gasper [Gas70, Gas72], consequently there is a continuous polynomial hypergroup J( a, ß) = (1, *(a,ß))

associated with {R~a,ß)} if and only if (a, ß) belongs to the set

E j = {(a,ß) : a 2': ß > -1 and ß2':-1/2 or a+ß2':O}

and there is a discrete polynomial hypergroup (No, *(a,ß)) associated with {R~a,ß)} if and only if (a, ß) belongs to the set

F j = {(a, ß) : a(a + 3)2(a + 5) 2': (a2 - 7a - 24)b2

where a=a+ß+1 and b=a-ß}.

The set Fj contains E j as a proper subset, so there can be no hope of an analog of Pontryagin duality for hypergroups. These include the exampIes discussed by Hirschman since the continuous polynomial hypergroup associated with {pJa)} is simply J(a - ~,a - ~).

2.4. Discrete polynomial hypergroups

We begin with a few observations. If </J is a character of a hypergroup (H, *) then II</Jlloo = SUPxEH I</J(x) I = 1. To see this, observe that </J(x)</J(y) = JH </Jd(8x * 8y ) immediately leads to II</JII~ ~ II</Jlloo, which implies II</Jlloo ~ 1. On the other hand, II</Jlloo ~ </J(e) = 1, if (H, *) is a discrete polynomial hypergroup associated with a family of orthogonal polynomials {Rn}. Since for n > 0, Rn is an unbounded continuous function, it follows that H must be a bounded set. A simple change of

110 A. Schwartz

variables allows us to ins ist that Hel with e = 1. H is necessarily Hermitian and commutative.

We recall that every family of orthogonal polynomials satisfies a recurrence relation

and there are sufficient conditions [Fav35, Sho36] that polynomials that satisfy such a relation are orthogonal with respect to a compactly supported non-negative measure. (Ql) has a fairly straightforward answer based on this last observation [Sch77].

Theorem 2.1. (No, *) is the discrete polynomial hypergroup associated with a family {Rn} of orthogonal polynomials if and only if

1. (On *ol)({k}) = 0 ifln - kl > l.

2. (On*Ol)({n+l}) >0.

Moreover, the polynomials may be normalized so that R 1(x) = X and the orthogonality measure is supported in I.

There are two types of responses to the harder quest ion (Q2). The first is a large collection of results such as Gasper's that have been obtained by workers in special functions. There are several general results based on the recurrence relation (2.6). The first resuIt we know of is due to R. Askey [Ask70], but it is subsumed by results due to Szwarc [Szw92a, Szw92b]; we state one from [Szw92a], but he has proved other theorems that are more widely applicable.

Theorem 2.2. 1f polynomials Pn satisfy

and

(i) an, ßn and an + "In are increasing sequences bn, an ~ 0),

(ii) an :::; "In for n E No,

then n+m

PnPm = L C~,mPk k=ln-ml

with C~,m ~ O.


This really is a discrete analogue to the Sturm-Liouville approach of the first lecture, since (2.6) may be reformulated as an eigenvalue difference relation:

'"Ynf:J.2 Rn(x) + (-yn - an)f:J.Rn(x) + (-yn + ßn)Rn(x) = xRn(x) ,

where f:J. 2Rn(x) = Rn+1(x) - 2Rn(x) + Rn-I(x) and f:J.Rn(x) = Rn(x)Rn-I (x). In fact, Szwarc's method is based on a discrete analogue of the hyperbolic Cauchy problem described in the first lecture. Markett has also devised a method generalizing the Riemann integration technique, which he can also use to obtain explicit linearization formulas [Mar94].

We refer the reader to the recent volume by Bloom and Heyer [BH95], which lists many more examples of discrete polynomial hypergroups (they use the term "polynomial hypergroups" for discrete polynomial hypergroups) .

2.5. Continuous polynomial hypergroups

The story here is a much simpler one and can be found in [CS90c, CMS92, CS95b]; complete answers to (Q3) and (Q4) are contained in the following theorems.

Theorem 2.3. If His an interval then (H, *) is a continuous polynomial hypergroup if and only if (H, *) is equivalent by means of a linear change of variables to a continuous Jacobi polynomial hypergroup J(a,ß) with (a, ß) E EJ .

Outline of proof. The "if" is essentially Gasper's result (see §). The "only if" is based on the idea of obtaining a second order linear differential equation for the characters, and then using an old result of Bochner [Boc29], which shows that there are only a few families of polynomials which satisfy such an equation. So we assume that (H, *) is a continuous polynomial hypergroup associated with an orthogonal family" {Rn}. To obtain the differential equation, begin with the relation

We now obtain a differential equation for Rn by following a path laid out in [Lev64]. Expansion of Rn(s) in a Taylor series around s = e and of Rn (r) around r = t yields

112 A. Schwartz

where Mk(8, t) = JH(r - t)kd(8s * 8t )(r). M1(8, t) and M2(8, t) can be computed explicitly in terms of R1 and R2 and then expressed in terms of 8 - e and (8 - e)2. For k > 0, Mk(8, t) has no 8 - e terms since Mk(8, t) = o(M2(8, t)) if k > 2. The coefficients of 8 - e on both sides of (2.7) are then equated to obtain

A(t)y"(t) + B(t)y'(t) = Any(t) (y = Rn, An = R~(e)).

• The same proof can be used to prove a result that weakens the con

dition that H be an interval:

Theorem 2.4. Suppose that (H, *) is a hypergroup where H is an infinite subset of lR and that for every n E No (H, *) has a character which is a polynomial Rn of degree n. Assume one of the following holds:

1. H is compact

2. {Rn} is orthogonal with respect to a positive Borel measure on H.

Then (H, *) is equivalent by means of a linear change of variables to a continuous Jacobi polynomial hypergroup J(a, ß) with (a, ß) E EJ .

3. Multivariate orthogonal polynomials and hypergroups

3.1. Multivariate orthogonal polynomials

The notion of orthogonal polynomials is somewhat more complicated for multivariate polynomials than it is for polynomials of one variable. In the latter case, the orthogonality measure determines the polynomials up to a constant multiple. In the multivariate case, the measure only determines a subspace Vn of polynomials (see below).

We first need some notation. We use following: Let 1/ E N = {l, 2, 3, ... } and let

dx = dXl dX2 ... dx",

k = (kl, k2 ,· .• , k,,), (k1, k2, ... , kv E No),

. Ikl = k1 + k2 + ... + k", x k = xk1 xk2 xkv

1 2··· ".

If p is a polynomial then deg p is the highest degree of a monomial that occurs in p with a non-zero coefficient. Let m E M(lRV ) and let H = supp m. Then m uniquely determines the subspaces

Vn = {p : degp = n and L xkp(x) dm(x) = 0 if Ikl < n}.

When v = 1, dirn Vn = 1 because the orthogonal polynomials in one variable are determined up to a multiplicative constant. In the general case dirn Vn > 1; for instance when v = 2, dirn Vn = n + 1. Thus the measure does not determine the individual orthogonal polynomials. One could construct the orthogonal polynomials of each degree by first selecting an order for the set of monomials {xk : Ikl = n}, and then applying the Gram-Schmidt orthogonalization process. But there are infinitely many possibilities not included among these n! choices, since the problem is really that of choosing an orthogonal basis for the vector space Vn with respect to the inner product (j, g) = JH f(x)g(x) dm(x). We say a set P of v-variable polynomials is algebraically complete if P is linearly independent and if for each n E No, {p E P : degp ::; n} spans the space of all polynomials of degree not exceeding n. When v = 1 an algebraically complete family is simply a set of polynomials that contains exactly one of each degree. We can obtain an algebraically complete family of polynomials by choosing a basis for each Vn and letting P be the union of those bases. In this case we say that P is a family of polynomials orthogonal with respect to m.

3.2. Continuous and discrete multivariate

polynomial hypergroups

Suppose that (H, *) is a hypergroup with H c lRv . We say that (H, *) is a continuous v-variable polynomial hypergroup if the character set of (H, *) contains a v-variable algebraically complete family of polynomials. Adefinition of discrete v-variable polynomial hypergroup can be readily formulated, but will not be needed in this lecture. The next few sections contain examples of multivariate continuous polynomial hypergroups.

3.3. Example. Product hypergroups; hypergroups on a square

It is always possible to obtain a hypergroup as a direct product of two hypergroups and in this way obtain continuous v-variable polynomial hypergroups for every v E N. In particular, if (H1, *d and (H2, *2) are hypergroups, then so is

114 A. Schwartz

where H = H 1 X H 2 . The product of point masses is given by the product measure

D(Xl,X2) * D(Yl,Y2) = (DXl *1 Dyl ) X (DX2 *2 Dy2 )·

Thus if (0., ß) = (al, ß1; a2, ß2) E EJ X EJ , J(o., ß) = J(a1, ß1) ® J(a2, ß2) is an example of a Hermitian continuous 2-variable polynomial hypergroup with characters

(-1< x,y < 1, n,m E No).

J(o.,ß) is a hypergroup on the square I x I with e = (1,1).

3.4. Example. Symmetrized product hypergroups

Let (a, ß) E EJ . Then J(a, ß) ® J(a, ß) can be symmetrized (c.f. [DR74]) with respect to the reflection r(x, y) = (y, x). The resulting hypergroup consists of measures on the triangle with vertices (1, 1), (1, -1), and (-1, -1). The polynomials

are characters, but this family is not algebraically complete since instead of the required two polynomials of degree one, there is only one. An algebraically complete family is obtained by the change of variables s = (x + y)/2, t = xy. The triangle is transformed to H = {(s, t) : 21sl- 1 ::; t ::; s2}. The result is a Hermitian continuous 2-variable polynomial hypergroup with e = (1,1).

3.5. Example. Disk polynomial hypergroups

Let'Y 2 0, z = reiB, and D = {z : Izl ::; I}. The polynomials of degree m + n

R'Y (z) = rln-mlei(n-m)B Rl'Y,ln-ml) (2r2 - 1) n,m nl\m,

(where n 1\ m is the minimum of n and m) are orthogonal on D with respect to the measure (1 - r2)'Yr dr de. Then

{R~,m(z) : n, mE No}

is the character set of a hypergroup on D which we denote Db) (see [Koo72, AT74, Kan76]). This is an example of a continuous 2-variable polynomial hypergroup that is non-Hermitian since ZV = z; e = 1.

Three Lectures on Hypergroups

3.6. Example. Hypergroups on a parabolic biangle

Let er ~ ß + 1/2 ~ 0, and let

R°,ß(x y) = R(o,ß+k+l/2)(2x _ 1) . Xk/2 R(ß,ß) (x-1/2y) n,k' n-k k'

115

Then there is a Hermitian continuous 2-variable continuous polynomial hypergroup (H, *) on the parabolic region

H = {(x, y) : y2 :S x :S I}

with characters R~:f with e = (1,1) [KS95].

3.7. Example. Hypergroups on a triangle

Let er ~ ß + 'Y + 1, ß ~ 'Y ~ -1/2, and let

R°,ß,i(X y) = R(o,ß+i+2k+1)(2x - 1) . xkR(ß,i) (x-1y - 1) n,k' n-k k'

Then there is a a Hermitian continuous 2-variable polynomial hypergroup (H, *) on the triangular region

H = {(x,y): y:S x:S I}

with characters R~:f'i with e = (1, 1) [KS95].

3.8. Example. Hypergroups on a simplex

The previous example can be generalized from the triangle to a kdimensional simplex

These are k-variable continuous polynomial hypergroups which are not products of lower dimensional hypergroups. The polynomials are defined recursively for k > 3, nl ~ n2 ~ ... nk by

and for k = 3 by

116 A. Schwanz

These are the characters of a Hermitian continuous k-variable polynomial hypergroup with e = (1,1, ... ,1) provided

See [KS95].

al 2 a2 + ... + ak+l + k - 1,

a2 2 a3 + ... + ak+l + k - 2,

ak-l 2 ak + ak+l + 1,

ak 2 ak+l,

ak+l 2 -~.

3.9. Classification of continuous 2-variable

polynomial hypergroups

The study of multivariate polynomial hypergroups is in a very early stage. For instance, there is to date some progress in the classification problem of the 2-variable continuous polynomial hypergroups, but there is nothing approaching the complete solution in the one-variable case.

We outline the 2-variable results in [CS95a]. Suppose (H, *) and (K, *) are continuous 2-variable polynomial hypergroups. If (H, *) and (K,*) are linearly equivalent, the sets Hand Kare connected by an affine transformation, so any attempt to classify hypergroups by means of the underlying set must first proceed by defining some canonical hypergroups. There are two such. We make the usual identifications of ]R2 with C and x = (x, y) = x + iy = z. (H, *) is a canonical Hermitian hypergroup if (x, y) I---t X and (x, y) I---t Y are characters. (H, *) is a canonical non-Hermitian hypergroup if z I---t z is a character. Canonical hypergroups have the following properties:

Theorem 3.1.

1. If (H, *) is a canonical H ermitian hypergroup, then

(a) The identity element of (H, *) is (1,1).

(b) He 12 .

(c) The first degree characters of (H, *) are (x, y) I---t X and (x,y) I---t y.

2. 1f (H, *) is a canonical non-Hermitian hypergroup, then

(a) The identity element 01 (H, *) is the complex number 1.

Three Lectures on Hypergroups

(b) ZV = z for every zEH.

(c) He D.

117

( d) The first degree characters 0 f (H, *) are z f-t z and z f-t z.

The following theorem shows that these two eanonieal types suffiee:

Theorem 3.2.

(i) 1f (H, *) is a Hermitian hypergroup that has exactly two distinct non-constant characters that are first degree polynomials, then it is linearly equivalent to a canonical Hermitian hypergroup.

(ii) 1f (H, *) is a non-Hermitian hypergroup that has exactly two distinct non-constant characters that are first degree polynomials, then it is linearly equivalent to a canonical non-Hermitian hypergroup.

The hypergroups J(o.,{3) and D(-y) play a special role in our diseussion. If (o.,{3) E E J X EJ , then J(o.,{3) is a Hermitian hypergroup, but it is not eanonieal. If we define the affine transformation A by

A(Xt, X2) = (R~Ol,ßll(xd, R~02,ß2)(X2))'

then Jc(o., (3) = A(J(o., (3)) is a eanonical Hermitian hypergroup defined on the rectangle [-'/'1,1] x [-1'2,1] with 1'v = (ßv + l)j(av + 1) for v = 1, 2. D(-y) is a eanonical non-Hermitian hypergroup defined on the disk D. Corollaries 3.5 and 3.6 show that there are no other eanonieal 2-variable eontinuous polynomial hypergroups defined on these sets. The next two theorems arrive at the same eonclusion with weaker geometrie hypotheses.

Theorem 3.3. Assume (H, *) is a compact canonical Hermitian 2-variable continuous polynomial hypergroup, then (H, *) = Jc(o.,{3) for some (o.,{3) E EJ X EJ if and only if H1 = H n (I x {I}) and H 2 = H n ({I} x 1) are infinite sets.

Remark 3.1. Example 3.4 shows the eondition in Theorem 3.3 eannot be weakened, sinee in that ease H1 and H2 each eontain only one point. Similarly, in Example 3.6, H2 = I, so it is an infinite set, but H1 eontains only a single point.

We need an additional definition in order to state the next result: Let x . y denote the usual inner product in R.2. If E C ]R2 we say that

118 A. Schwartz

Xo is a 2-dimensional accumulation point of E if Xo is an accumulation point of E, and the only a E ]R2 satisfying

a·(x-xo)=o(llx-xoll) (x-+xo with XEE)

is a = O. The identity element is a 2-dimensional accumulation point of H in every one of the examples except the symmetrized product hypergroups (§3.4).

Theorem 3.4. If (H, *) is a canonical non-Hermitian 2-variable continuous polynomial hypergroup, then (H, *) = D('y) for some "I ~ 0 if and only if the identity element 1 is a 2-dimensional accumulation point of H, and {z EH: Izi = 1} contains at least seven points.

The following two obvious corollaries describe the canonical 2-variable hypergroups which are maximal regarding the underlying set.

Corollary 3.5. If R is a rectangle with sides parallel to the coordinate axes and (R, *) is a canonical Hermitian 2-variable continuous polynomial hypergroup, then (R,*) = Jc(a,ß) for some (a,ß) = (0:1, ß1; 0:2, ß2) E E J X E J . In particular, if R = [-"11,1] X [-"12,1], then ßv = "Iv(O:v + 1) - 1 for v = 1, 2.

Corollary 3.6 If(D, *) is a canonical non-Hermitian 2-variable continuous polynomial hypergroup, then (D, *) = D('y) for some "I> O.

Examples 3.4, 3.6 and 3.7 show that Theorem 3.3 is not exhaustive of the Hermitian continuous 2-variable polynomial hypergroups; we do not know at this time of any examples that show that Theorem 3.4 is not exhaustive for the non-Hermitian case.

The proof of Theorems 3.3 and 3.4 rely heavily on the idea that the characters of certain continuous 2-variable polynomial hypergroups satisfy a pair of second-order linear partial differential equations.

Theorem 3.7. Let (H, *) be a 2-variable continuous polynomial hypergroup and assume that e is a 2-dimensional accumulation point of H. Then there is a pair of second-order linear partial differential operators LI and L 2 such that for every character cjJ of (H, *) we have

The derivation of these equations uses the same ideas as in the first lecture. We believe that the corresponding equations in the higher dimensions will provide the key to identify many hypergroups, but Example 3.4 shows that there are hypergroups to which this theorem does not apply.

4. Research questions

We gather here several questions that arise from the material in the lectures:

Question 1. Find necessary and sujJicient conditions that the characters of a hypergroup are eigenfunctions of a Sturm-Liouville problem. The one-dimensional hypergroups that 1 know of are all, in fact, of Sturm-Liouville type. Do there exist one-dimensional hypergroups which are not of Sturm-Liouville type? 1s there a hypergroup (J, *) which is not equivalent to a Jacobi type hypergroup?

Question 2. Theorems 2.3 and 2.4 give rise to the following question: To what extent can the hypothesis that there be a polynomial of every degree among the characters of (H, *) be weakened? Perhaps this can be done if "linear change of variables" is replaced by "chanre of variables". For instance, the even ultraspherical polynomials {lR~~ (x)} are the characters of a hypergroup on [0,1]. But by [Sze67, (4.1.5)J

lR~~)(x) = R~<>'-~)(2x2 - 1). 1s it possible that a family of polynomials other than the Jacobi polynomials disguised by a change of variables can appear as all or some characters of a hypergroup?

Question 3. There is still a "mousehole" left in Theorem 2.4. 1s there a hypergroup with polynomial characters which is not equivalent to any J(a, ß)? For the answer to be "yes" it would be necessary that H be a bounded non-compact set and that the polynomials not be orthogonal. Haar measure would necessarily be unbounded.

Question 4. Suppose a Sturm-Liouville type hypergroup (H, *) on a compact interval H has a dual structure which is also a hypergroup. This is the same as simply requiring that any two eigenfunctions of a Sturm-Liouville problem can be expressed as a finite linear combination of eigenfunctions with non-negative coejJicients. Does this imply that (H, *) is equivalent to J(a, ß) for some (a, ß) E E]?

120 A. Schwanz

Question 5. There are polynomials besides the Jacobi polynomials which have positive product formulas and give rise to Banach algebras of measures which are not hypergroups. This class needs funher investigation. Such polynomials include the generalized Chebyshev polynomials fLai80j and the q-ultraspherical polynomials fAI83}.

Question 6. Which plane regions support canonical 2-variable continuous polynomial hypergroups? In particular, are there any canonical nonHermitian 2-variable continuous polynomial hypergroups asidefrom Db) and subgroups of the circle?

Question 7. Do all 2-variable continuous polynomial hypergroups have characters which are related to Jacobi polynomials? This is probably the case when e is a 2-dimensional accumulation point of H because then the polynomials must satisfy differential equations (Theorem 3.7). We see from the example of symmetrized product hypergroups that the charaeters are related to Jacobi polynomials even when e is not a 2-dimensional accumulation point of H.

Question 8. Given a measure m E M(][~2) can there be more than one 2-variable continuous polynomial hypergroup with Haar measure m; that is, is it ever possible to seleet two different families of polynomials orthogonal with respect to m such that there are hypergroups with each family of polynomials as charaeters?

5. Guide to the references

As mentioned in the introduetion, I am including a more extensive bibliography than is required for these notes. The listing below keys the bibliographie items by topie. For a mueh more extensive list of referenees the reader should eonsult the book of Bloom and Heyer [BH95] or Litvinov's survey [Lit87].

Reference works

[BH95], [Edw67], [EMOT53], [GR90], [Hey84a], [Hey86], [Hey89], [Hey91], [Jew75], [Rud62], [Rud74],[Sze67], [Wat66]

Hypergroups (and related systems) in general

[BH95], [BK92], [CGS95], [CS92], [DR74], [Dun73], [Geb89], [Geb95], [Jew75], [Lev64], [Lit87], [Ros77], [Ros78], [Ros95], [Sch74], [Seh77], [Seh88], [Spe78], [Zeu89], [Zeu91]


Ultraspherical and Jacobi polynomials

[AF69], [Ask74], [CMS91], [CS77], [CS79], [CS89], [CS90e], [CS95b], [Gas70], [Gas72], [Hir56a], [Hir56b], [Koo72], [Koo74a]

Other one-variable polynomials

[AAA84], [AI83], [AKR86], [Ask70], [Boe29], [DR74], [Fav35], [Koo78], [Lai80], [Las83], [L091], [Mar94], [Rah86], [Sho36], [Soa91], [Szw92a], [Szw92b], [Voi90]

Other special functions

[CMS93], [FK73], [FK79], [Fle72], [Koo75a], [Mar89], [Per86], [Seh71]

Disk polynomials

[AT74], [BG90], [BG91], [BG92], [GS96], [HK93], [Kan76], [Kan85]

Multivariate polynomials

[CS95a], [KMT91], [Koo74b], [Koo75b], [KS67], [KS95], [Mae90], [Tra91]

Differential equations

[AT79], [Che72], [CMS92], [CS90a], [CS90b]

Probability on hypergroups

[BG90], [BG92], [GG87], [Hey84a], [Hey86], [Hey84b], [Hey91], [Voi90]

Miscellaneous

[Bru87], [Dun66], [Mie51]

[AAA84] W. Al-Salam, W. R. Allaway, and R. Askey. Sieved ultraspherieal polynomials. Trans. Amer. Math. Soc., 284:41-54, 1984.

[AF69] R. Askey and J. Fiteh. Integral representations for Jaeobi polynomials and some applieations. J. Math. Anal. Appl., 26:411-437, 1969.

122 A. Schwanz

[AI83] R. Askey and M. E. H. Ismail. A generalization of ultraspherical polynomials. In P. Erdos, editor, Studies in Pure Mathematies, pages 55-78, Boston, 1983. Birkhauser.

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Email: [email protected]; Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis, MO 63121

Harmonie Analysis and Funetional

Equations

H enrik StetktET

1. Introduction

Functional equations oeeur in many parts of mathematics, also in harmonie analysis. As an example we mention that the eomplex exponential function "( : x -+ exp (~x) for any ~ E R is a solution of Cauehy's functional equation

"((x + y) = "((x)"((y), X,yE R (1)

More generally, the so-ealled spherieal functions play the role in harmonie analysis on homogeneous spaees that the trigonometrie polynomials do in Fourier analysis. Also, the spherical functions ean be eharacterized as solutions of eertain functional equations. We will in this leeture study a special type of functional equations and point out some of its relations to (1) spherieal funetions, (2) mean value properties, and (3) addition formulas.

The general set up. Let G be an abelian topologieal group, and let K be a eompaet topologieal group aeting as a topologieal transformation group on G. We write k· x for the action of k E K on x E G. We assurne that K aets by automorphisms of G, so that the map 9 -+ k . 9 of G into Gis an automorphism for eaeh fixed k E K. The action of K on G gives rise to an action of K on the functions on G: If! is a function on G and k E K we let (k· J)(x) := !(k-1 • x), xE G. Finally we let dk denote the normalized Haar measure on the eompaet group K.

As an example we mention that the integralover Z2 = {±1} is fz2h(k)dk = (h(+l) + h(-1))/2. In this example (-1)· x = ax for all x E G, where a : G -+ Gis a eontinuous automorphism of G such that a2 = I. The classical instanee is a = -I.

We shall examine eertain functional equations of the general form

N 1 J(x + k· y)dk = ~91(X)hl(Y)' X,yE G, (2)

132 H. Stetkcer

where the functions J,91, ... ,9N,h1, ... ,hN E C(C) are to be determined.

For later use we note the following result:

Theorem A. Let91, ... ,9N,hl, ... ,hN E C(C). IJthe equation (2) has a solution for J E C( C) then

N t; L 91(X + k· y)dk h1(z)

N

= t;91(X) fKh1(Y+k.z)dk Jor alt x,y,z E C. (3)

Proof. For any J E C(C) and x, y, z E C we consider the iterated integral

fK L J(x + k· y + k'· z)dk'dk = L fK J(x + k· [y + (k-1k') . z])dk'dk.

When we on the right hand side first introduce k1 = k-1 k' as variable instead of k' and then change order of integration we get the identity

L fK J(x + k· y + k' . z)dk'dk = L fK J(x + k· [y + k1 . z])dkdk1.

Applying the formula (2) to this identity we obtain (3). • Theorems 3, 6 and 7 have only appeared in preprint form, and The

orem 5 is new.

2. Spherical functions

An example of the functional equation (2) is d'Alembert's functional equation

fK fjJ(x + k . y)dk = fjJ(x)fjJ(y), x,y E C, (4)

where fjJ E C( C) is the unknown. For K = {I} it reduces to Cauchy's functional equation (1). A particular case for K = Z2 is the functional equation

fjJ(x + y) ; fjJ(x - y) = fjJ(x)fjJ(y), x,yE R, (5)

Harmonie Analysis and Functional Equations 133

that d'Alembert studied in connection with his investigations of the wave equation [4]. His functional equation (5) is also called the eosine equation, because its non-zero solutions are the functions of the form cfJ(x) = cos(>.x) , >. E C.

Another interesting instance is the case of K = 80(2) acting in the natural way on G = R 2 = C. Here d'Alembert's functional equation takes the form

1 r27f 27r Jo cfJ (x + ei()y) dB = cfJ(x)cfJ(y), x,y E C.

It is known that the non-zero (continuous) solutions are described by certain complex Bessel functions (See Proposition IV.2.7 of [11]).

Definition. A K-spherical function on G is a solution cfJ E C(G), cfJ i= 0, of d'Alembert's functional equation (4).

In the standard terminology our K -special functions are the spherical functions on the semidirect product G X s K with respect to the subgroup K. Note that cfJ = 1 is a K-spherical function.

Proposition 1. Any function of the form fK k·'Y dk, where'Y: G ---; C* is a eontinuous homomorphism, is K -spherieal.

If K is finite then a eontinuous homomorphism "10 : G ---; C* for which fK k· "10 dk = fK k· "I dk has the form "10 = ko· "I for some ko E K.

Proof. Let cfJ := fK k·'Ydk. Noting that k·'Y is a homomorphism because "I is, we get by a change of order of integration that

L cfJ(x + k1 . y)dk1 = L L (k . "I) (x + k1 . y)dk dk1

= L L (k· 'Y)(x)(k· 'Y)(k1 · y)dkdk1

= L (k· 'Y)(x) {L (k· 'Y)(k1 . Y)dk1} dk

= L (k· "I) (x) {L 'Y((k-1k1) . Y)dk1} dk.

The translation invariance of the Haar measure reduces this to

L cfJ(x + k1 . y)dk1 = L (k . "()(x) {L 'Y(k1 . y)dk1 } dk

cfJ(x) L 'Y(k1 . y)dk1 .

134 H. Stetkcer

By the invarianee of the Haar measure und er involution we get

1 cjJ(x + k1 . y)dk1 = cjJ(x) 1 (k11 . "()(y)dk1

= cjJ(x) 1 (k1 . "()(y)dk1 = cjJ(x)cjJ(y).

The last statement of the proposition is a eonsequenee of the faet that the set of homomorphisms of G into C* is linearly independent in the vector spaee of all eomplex valued functions on G. (See for example Lemma 29.41 of [12] for a proof.) •

The following result is derived in [17] by classical harmonie analysis.

Theorem 1. If cjJ is a K-spherical function on G = Rm x Tn then there exists a continuous homomorphism "( : G --7 C* = C\ {O} such that cjJ = J K k . "( dk.

For general G very little is known about the solutions of d'Alembert's functional equation (4). However, for K = Z2 its solution set was found by Kannappan [13] for a = -1, and by Baker [6] for any automorphism a of order 2.

Theorem 2. Let a : G --7 G be a continuous automorphism for G such that a2 = I. Any continuous solution cjJ =I 0 of d'Alembert's functional equation

cjJ(x + y) ~ cjJ(x + ay) = cjJ(x)cjJ(y), x,y E G,

may be written in the form cjJ = b + "( 0 a)/2, where "( : G --7 C* is a continuous homomorphism.

"( is essentially unique: If "(1 : G --7 C* is a homomorphism and b1 + "(1 0 a)/2 = b + "( 0 a)/2 then either "(1 = "( or "(1 = "( 0 a.

Theorem 2 says that the solutions of d'Alembert's functional equation ean be expressed by homomorphisms just like the eosine function ean be expressed by exponentials via Euler's formulas. Inspired by this, the idea of the proof of Theorem 2 is, for given cjJ, to try to find a eorresponding homomorphism "( as a linear eombination of cjJ and translates of cjJ. More preeisely, it turns out to be a good idea to try

"((x) = cjJ(x) + k(</J(x + xo) - cjJ(x + axo)), xE G,


where k E C and Xo E Gare to be chosen judiciously. For details the reader can consult the proof of Theorem III.1 of [19].

In Theorems 1 and 2 we observe that the basic building blocks of harmonie analysis, i.e. the continuous homomorphisms "( : G --t C·, play a decisive role.

We have not assumed above that the solutions were bounded. The special case of bounded solutions can be handled using classical harmonie analysis. See Chojnacki [7], [8], Badora [5] and Stetkcer [15].

d'Alembert's classical functional equation (5) was generalized in 1919 by Wilson [20] to (f(x + y) + f(x - y))/2 = f(x)cjJ(y) , x,y E R. We generalize it still furt her to

L f(x + k . y)dk = f(x)cjJ(y), x,yE G, (6)

in which there are 2 unknown functions fand cjJ instead of only one. Actually, this new functional equation (6) also occurs in the theory of spherical functions (see, e.g. Proposition IV.2.4 of [11]). The most general result known for abelian groups is the following which is due to Aczel, Chung and Ng [1] for the case a = -1 (see also [16] and [19]).

Theorem 3. Let a : G --t G be a continuous automorphism of order 2. Let f, cjJ E C( G) satisfy Wilson's functional equation

f(x + y) ~f(x + ay) = f(x)cjJ(y), x,y E G. (7)

(a) 1f f i- 0 then cjJ has the form cjJ = b+"(oa)/2 for some continuous homomorphism "( : G --t C·. We assume from now on that cjJ has this form.

(b) 1f"( i- "( 0 athen the set of solutions f of Wilson's functional equation (7) consists of all functions of the form

where c and d are complex constants.

(c) 1f"( = "( 0 athen the set of solutions f of Wilson's functional equation (7) consists of all functions of the form f = "((c + a-), where c is a complex constant and a- : G --t C is a continuous additive map such that a- 0 a = -a-.

136 H. StetkceT

Proof. (a) Theorem A gives here that <jJ is a K -spherical function so that Theorem 2 applies.

(b) + (c): It is easy to see that the formulas define solutions, so it is left to prove that any solution f has one of the described forms. Letting f± := (J ± f 0 a)/2 we find that

J±(x + y) ~ f±(x + r7Y) = f±(x)<jJ(y), x, Y E G.

Taking x = 0 we find f+ = f+(O)<jJ, which is the first term of funder (b) as weH as (c). We interchange x and y in

to obtain

f-(x + y) + f-(x + r7Y) = f-(x)<jJ(y), 2

f-(x + y) - f-(x + r7Y) = f-(y)<jJ(x), 2

Adding these two identities yields

f-(x + y) = f-(x)<jJ(y) + <jJ(x)f-(y) ,

x,y E G,

x,y E G.

x,y E G.

(c) If "( = "( 0 aso that "( = "( 0 r7 = <jJ then we get from (8) that

x,y E G,

which proves (c).

(8)

(b) We will view (8) as a special case of the functional equation (2) by writing it in the form IK f-(x + k· y)dk = f _(x)<jJ(y) + <jJ(x)f _(y + z), where K = {I} is the 1 point group. Applying Theorem A to this special case of (2) we get the identity

f-(x + y)<jJ(z) + <jJ(x + Y)f-(z) = f-(x)<jJ(y + z) + <jJ(x)f-(y + z).

We substitute the express ions for f-(x + y) and f-(Y + z) from (8) into the identity and find after cancellation of some terms that

f-(x)[<jJ(y + z) - <jJ(y)<jJ(z)] = f-(z) [<jJ(x + y) - <jJ(x)<jJ(y)].

Substituting

<jJ(x + y) - <jJ(x)<jJ(y) = "((x) ~ "((ax) "((y) -2 "((ay)

we get (b). •

Harmonie Analysis and Funetional Equations 137

3. Mean value properties

A special case of Wilson's generalization (6) of d'Alembert's functional equation comes ab out for cP = 1 where the functional equation reduces to

[f(X + k· y)dk = f(x), x,y E G. (9)

To put things into perspective, one should observe that replacing x by (~+ ry)/2 and y by (~ - ry)/2 in the special case

f(x + y) ; f(x - y) = f(x), x,y E R,

of (9) we get Jensen's classical functional equation

f(~) ; f(ry) = f (~; 77) .

If the equality sign is replaced by 2:: we have Jensen's inequality for convex functions. As a coroHary of Theorem 3 we solve the foHowing version of Jensen's functional equation:

Theorem 4. Let (7 : G ---t G be a eontinuous automorphism of order 2. The set of solutions f E C(G) of Jensen's funetional equation

f(x + y) ~ f(x + (7Y) = f(x), x,YEG, (10)

eonsists of the functions of the form f = e + a-, where e is a eomplex eonstant and a- : G ---t C is a eontinuous additive map sueh that a- 0(7 = -a-.

The functional equation (9) describes a mean value property of f. In the example of K = 80(2), acting in the natural way on G = R 2 = C, the functional equation says that

r27r dU Jo f (x + eiOy) 211" = f(x), x,yE C,

Le. the value of f at the center of any circle is the mean of its values on the circle. As is weH known this characterizes the harmonie functions.

138 H. Stetka;r

Functions having the mean value property over other geometrie figures than cirdes, e.g. vertices of regular polygons or rectangles in R 2,

have been studied by Aczel et al. [3], Chung et al. [9] and in a more general situation by Stetkrer [18].

The functional equation corresponding to regular N-gons is

1 N-l

N L fex + wny) = fex), n=O

x,y E C,

where w = exp(2ni/N). The equation is dearly of the form (9) above with K = ZN. Its set of solutions is span {1,Z,oo.,zN-l,z,oo.,zN-l} (see, for example Theorem 111.4 of [18]).

The reet angular functional equation which reads

f(Xl + Yl, X2 + Y2) + f(Xl + Yl, X2 - Y2) + f(Xl - Yl, X2 + Y2)

+ f(Xl - Yl, X2 - Y2) = 4f(Xl' X2) for (Xl, X2), (Yl, Y2) E R 2 ,

(11)

falls into our framework with K = Z2 X Z2 acting coordinatewise on R 2 • Its set of solutions are all functions of the form f(Xl, X2) = a12Xlx2 +alxl + a2X2 +ao, where a12, al, a2, ao are complex constants [3]. A generalization of it is the functional equation

r 00. r f(Xl+kl'Yl,oo.,xn+kn'Yn)dkloo.dkn=f(Xl,oo.,Xn), lZ2 lZ2

(12)

where K = Z2 X ..• X Z2 acts coordinatewise on G = Gl X ... x Gn .

Let us for each i = 1, ... , n write the action of -1 E Z2 = {±1} on Gi as (-1) . Xi = aiXi so that ai : Gi ~ Gi is a continuous involution. For ai = -I, i = 1, ... , n, the following theorem is derived by a long induction on n in [9]. Our result generalizes not just the formulas for the solutions of the reet angular functional equations (11) and (12) of [3] and [9] but also the special case of Jensen's functional equation from Theorem 4 above.

Theorem 5. The set of solutions of (12) consists of the linear span of the functions 0 f the form (Xl, ... , Xn) ~ ap1",pl (Xp1 , ... , Xp1 ) with 1 ::; Pl < ... < PI ::; n,O ::; I ::; n, where ap1'''PI : Gp1 x ... X Gp1 ~ C ranges over the continuous multi-additive functions satisfying

for j = 1, ... , I. If I = 0 the function shall be interpreted as the constant function 1.

Harmonie Analysis and Funetional Equations 139

Proof. It is elementary to verify that any function of the deseribed form is a solution of (12), so it is left to show that any solution f has this form.

Let K = Z2 x··· X Z2. It is known from the theory of representations of eompact groups (see, e.g., Lemma IV. 1.9 of [11]) that the deeomposition of a function into an even and an odd part has an extension to the present set up: Any f E C(G) may be written as f = LXEf< f x where

f x = fK k . fX(k) dk has the property k . f x = x(k)fx , k E K. As is easy to see, f x satisfies the functional equation (12), so replaeing f by f x we may from now on assurne that f satisfies the invarianee eondition k· f = x(k)f, k E K, for some homomorphism X : Z2 x '" X Z2 -t T. Any sueh homomorphism has the form

where Xl : Z2 -t T is a homomorphism for each i = 1,2, ... , n. It follows from (12) that f as a function of its i'th variable is a solution

of Jensen's functional equation (10), and from the invariance property that ki . f = Xi(ki)f· Prom Theorem 1 we get that f does not depend on its i'th variable if Xi = 1, and that f is additive and satisfies (13) in its i'th variable if Xi =J 1. •

4. Addition formulas

The topic that I want to diseuss last is the addition formulas of sine and eosine. We will show that these formulas are closely related.

If K = {I} and G = R then the pair f = sin and 9 = eos constitute a solution of the functional equation

1 f(x + k· y)dk = f(x)g(y) + g(x)f(y), x,y E G. (15)

This is just the addition formula for sine. The addition formula for eosine is

l g(x + k . y)dk = g(x)g(y) - J(x)J(y), x,yE G. (16)

Theorem 6. IJ J, 9 E C( G) constitute a solution oJ the Junetional equation (15) and J =J 0, then there exists a constant K, E C such that

l g(x + k . y)dk = g(x)g(y) + K,f(x)f(y), x,y E G.

140 H. StetktEr

Furthermore both fand gare K -invariant in the sense that f(k·x) = f(x) and g(k· x) = g(x) for all k E K and xE G.

In other words, g must satisfy the addition formula for eosine. For K = {I} the result can be found as Proposition 13.2 of [2].

Proof. We first prove the statement on K-invariance. Let ko E K. Replacing y by ko . y in (15) we get by the translation invariance of the Haar measure on K that f(x)g(ko·y)+g(x)f(ko·Y) = f(x)g(y)+g(x)f(y). The statement follows if fand gare linearly independent. If they are not then g = J.lf for so me J.l E C and the functional equation reduces to IK f(x + k· y)dk = 2J.lf(x)f(y)· Now, J.l i= 0, because J.l = 0 implies that f = 0 contradicting the assumption. Taking Xo E G such that f(xo) i= 0 we find that

f(y) = 2J.l}(xo) fK f(xo + k . y)dk.

This formula shows that f is K-invariant. Hence so is g = J.lf. Observe for use below that

fKg(y+k.x)dk= fKg(k-1.y+X)dk= Lg(X+k.y)dk

by the invariance of the Haar measure under the change of variables k ---t k- 1 .

Theorem A applied to the functional equation shows that

f(x) Lg(Y+k,z)dk = g(z) L f(y+k·z)dk = [J(x)g(y)+g(x)f(y)]g(z),

so that f(x)[JK g(y+k· z)dk - g(y)g(z)] = g(x)f(y)g(z). The right hand side is unchanged if x and z change places, hence so is the left hand side, i.e. f(X)[JK g(y+ k· z)dk - g(y)g(z)] = f(z)[IK g(y+k· z)dk - g(y)g(x)]. Applying the observation above to the right hand side we find that

f(x) [!Kg(y + k· z)dk - g(y)g(z)]

=f(z) [Lg(x+k.y)dk-g(x)g(y)]. (17)

Now, f(zo) i= 0 for so me Zo E G. Putting z = Zo and dividing through by f(zo) we see that there is a function 'ljJ : G ---t C such that

L g(x + k . y)dk - g(x)g(y) = f(x)'ljJ(y), x,y E G.

Introducing this expression into (17) gives that f(x)f(y)'ljJ(z) f(z)f(x)'ljJ(y) from which we see that 'ljJ is a constant multiple of f.

•


Theorem 7. If f, gE C(G) eonstitute a solution of the funetional equation (16) then there exists a eonstant n E C sueh that

fK f(x + k 0 y)dk = f(x)g(y) + g(x)f(y) + nf(x)f(y), x,y EGo

Furthermore both fand gare K -invariant in the sense that f(kox) = f(x) and g(k 0 x) = g(x) for all k E K and x E Go

Proof. The proof of the K -invarianee of fand 9 goes along the same lines as in Theorem 6, so we skip ito

If fand 9 satisfy (16), Theorem A teIls us that

[g(x + k 0 y)dkg(z) - [f(x + k 0 y)dkf(z)

=g(x) [g(Y+koz)dk-f(x) [f(Y+koz)dk

so that by (16)

[g(x)g(y) - f(x)f(y)]g(z) - [f(x + k 0 y)dkf(z)

= g(x)[g(y)g(z) - f(y)f(z)]- f(x) [f(Y + k 0 z)dk

or

f(x) [f(Y+koz)dk- f(x)f(y)g(z) = f(z) [f(x+koY)dk-g(x)f(y)f(z)o

Subtraet J(x)g(y)J(z) from both sides to get

J(x) [[ J(y + k 0 z)dk - J(y)g(z) - g(y)J(z)]

= J(z) [[ J(x + k 0 y)dk - J(x)g(y) - g(x)J(y)] 0 (18)

If J = 0 the eonclusion of the theorem is trivially trueo Thus we ean assurne that there is Zo E G at whieh f(zo) i= 00 Now let

c/>(x, y) := [f(x + k 0 y)dk - f(x)g(y) - g(x)f(y)o

Using z = Zo the identity (18) now reads f(x)c/>(y, zo) = f(zo)c/>(x, y), x, y E G, so that

c/>(x, y) = f(x)'IjJ(y), where 'IjJ(y) := c/>(y, zo)/ f(zo)o (19)

142 H. Stetka;r

Using z Zo the identity (18) now reads f(x)c/>(y, zo) f(zo)c/>(x, y), x, y E C, so that

c/>(x, y) = f(x)'l/J(Y), where 'l/J(y) := c/>(y, zo)j f(zo). (19)

Substitute (19) into (18) to get f(zo)f(x)'l/J(Y) = f(x)f(y)'l/J(zo) or f(zo)'l/J(Y) = f(y)'l/J(zo). Thus 'l/J(y) = af(y), and c/>(x, y) = af(x)f(y), proving the theorem. •

Much more should be known about the addition formulas (15) and (16) for a general action of a compact group K, but to the best of my knowledge only the cases of K = {I} and K = Z2 have been solved completely (see [10] and [14]). The solutions involve group homomorphisms 'Y : C -t C* so harmonic analysis should intervene in the general set up.

I hope in this lecture to have convinced you that there are interesting and important relations between the basic building blocks of harmonic analysis and functional equations, and that the topic is far from being completely explored.

References

[1] Aczel, J., Chung, J. K. and Ng, C. T.: Symmetrie seeond differences in product form on groups. Topics in mathematical analysis (pp. 1-22) edited by Th. M. Rassias. Ser. Pure Math., 11, World Scientific Publ. Co., Teaneck, NJ, 1989.

[2] Aczel, J. and Dhombres, J.: Functional equations in several variables. Cambridge University Press CambridgejNew YorkjNew Rochellej MelbournejSydney 1989.

[3] Aczel, J., Haruki, H., McKiernan, M. A. and Sakovic, G. N.: General and Regular Solutions of Functional Equations Characterizing Harmonie Polynomials. Aequationes Math. 1 (1968), 37-53.

[4] d'Alembert, J., Reeherehes sur la eourbe que forme une eorde tendue mise en vibration, 1-JI. Hist. Acad. Berlin (1747) 214-249.

[5] Badora, R,On a joint generalization of Cauehy's and d'Alembert's funetional equations. Aequationes Math. 43 (1992), 72-89.

[6] Baker, J. A., The stability of the eosine equation. Proc. Amer. Math. Soc. 80 (1980), 411-416.

[7] Chojnacki, W., Fonctions cosinus hilbertiennes bornees dans les groupes eommutatifs loealement eompaets. Compositio Math. 57 (1986), 15-60.


[8] Chojnacki, W., On some funetional equations generalizing Cauehy's and d'Alembert's funetional equations. Colloq. Math. 55 (1988),169-178.

[9] Chung, J. K. (Zhong Jukang), Ebanks, B. R., Ng, C. T., Sahoo, P. K. and Zeng, W. B., On generalized rectangular and rhombie funetional equations. PubI. Math. (Debrecen) 47 (1995), 249-270.

[10] Chung, J. K., Kannappan, PI. and Ng, C. T., On two trigonometrie funetional equations. Mathematics Reports Toyama University 11 (1988), 153-165.

[11] Helgason, S.: "Groups and Geometrie Analysis." Academic Press, Inc., Orlando - San Diego - San Francisco - New York- London -Toronto - Montreal - Sydney - Tokyo - Säo Paulo 1984.

[12] Hewitt, E. and Ross, K. A.: "Abstract Harmonie Analysis II". Springer-Verlag. Berlin-Heidelberg-New York 1970.

[13] Kannappan, PI., The functional equation f(xy) + f(xy-l) = 2f(x)f(y) for groups. Proc. Amer. Math. Soc. 19 (1968), 69-74.

[14] Poulsen, T. and Stetkrer, H., Funetional equations on abelian groups with involution. Preprint Series 1995 No 16, Matematisk Institut, Aarhus University, Denmark, pp. 1-23.

[15] Stetkrer, H., D'Alembert's equation and spherieal funetions. Aequationes Math. 48 (1994), 220-227.

[16] Stetkrer, H., Wilson's Functional Equations on Groups. Aequationes Math. 49 (1995), 252-275.

[17] Stetkrer, H., Funetional Equations and Spherieal Funetions. Preprint Series 1994 No 18, Matematisk Institut, Aarhus University, Denmark, pp. 1-28.

[18] Stetkrer, H., Wilson's funetional equation on C. Preprint Series 1995 No 1, Matematisk Institut, Aarhus University, Denmark. pp. 1-15. Accepted for publication by Aequationes Math.

[19] Stetkrer, H., Functional equations on abelian groups with involution. Accepted for publication by Aequationes Math.

[20] Wilson, W. H., On eertain related junetional equations. BuH. Amer. Math. Soc. 26 (1919), 300-312.

E-mail: [email protected]; Department of Mathematics, Aarhus University, Ny Munkegade, DK 8000 Aarhus C, Denmark

Actions of Finite Hypergroups and

Examples

v. S. Sunder and N. J. Wildberger

Abstract

This paper is an introduction to the theory of actions of finite hypergroups, particularly commutative ones. We present so me basic facts concerning actions and then proceed to dassify irreducible *-actions of hypergroups of order two, the dass and character hypergroups of 3 3 and of the Golden hypergroup -which arises from the pentagon when viewed as a strongly regular graph.

1. Introduction

This paper will try to illustrate the idea of an action of a finite hypergroup Je = {co, ... , cn } on a finite set X through the investigation of some specific examples. The study of actions of objects related to hypergroups was initiated in [11] in the study of invariants associated to indusions of II1 factors. We will adapt the not ion studied there to finite hypergroups in the sense of Dunkl [4], Jewett [6] and Spector [9]; see also [2], [7], [8].

An action is a special type of representation; one where the representing matrices are an column stochastic, that is with non-negative real entries and column sums 1. If the action preserves the hypergroup involution *, that is if ci is mapped to the adjoint of Ci for an i, then we say we have a *-action. In this case the representing matrices will be doubly stochastic. This is the most natural situation from some points of view, and we will be concentrating on it.

An action can be thought of geometrically as an assignment to each Ci of an affine map of some simplex to itself. A *-action is a special type of action for which in particular the centroid of the simplex is left fixed by an Ci.

An action or *-action is irreducible if no face of the simplex is left stable by all Ci. This definition has no counterpart in representation theory; thus it is possible for even a commutative hypergroup to have irreducible *-actions of dimension larger than one. However one of the fundamental facts, as given by Theorem 5, is that this dimension is

146 V.S. Sunder and N.J. Wildberger

bounded by the total weight w(K) of the hypergroup. After giving the main definitions of hypergroups, actions and *

actions we discuss some general families of examples arising from quotient hypergroups and dass hypergroups of finite groups. We then prove the main inequality mentioned above for the dimension of an irreducible *-action of a commutative hypergroup and discuss some related results. We introduce the not ion of a maximal action and state a relation between maximal *-actions and association schemes.

The last section is devoted to the dassification of irreducible *-actions of certain commutative hypergroups. In particular, we consider the general hypergroup with two elements, the dass and character hypergroups of the symmetrie group S3, and the so-called Golden hypergroup.

Some of the results stated here are proved in a more general context in [12]. The present paper is intended to be an introduction to this cirde of ideas; there are a number of obvious questions which we do not at present know how to answer. Among these are the following.

Problem 1. Given a commutative hypergroup K, does K admit only a finite number of irreducible *-actions?

Problem 2. How can one determine these irreducible *-actions? In particular, is there any 'internal' description of them?

Problem 3. Which hypergroups admit a maximal *-action, that is, one where the dimension is exactly w(K)? (A partial answer is given in Theorem 7).

Problem 4. The dass hypergroup K(G) for any finite group always admits a maximal *-action. Is this unique? How about if G is simple?

2. Definitions of hypergroups and actions

Definition 1. A finite hypergroup is a pair (K, A) where A is a *algebra with unit Co over C and K = {co, Cl, ... ,Cn } is a subset of A satisfying

(Al) K is a basis of A (A2) K* = K (A3) The structure constants nt E C defined by

CiCj = L:ntck k

Actions 0/ Finite Hypergroups and Examples

satisfy the conditions

C; = Cj ~ n?j > 0

c; i= Cj ~ n?j = 0.

147

K is called Hermitian if cr = Ci for all i and commutative if CiCj = CjCi for all i, j. Hermitian hypergroups are always commutative. In this paper, we will consider commutative hypergroups primarily.

If we replace axiom (A4) with

(A4') n~j E JE.

then we say K is a signed hypergroup.

Definition 2. The weight of an element Ci E K is W(Ci) = (n~J-l where Cj = er. The total weight of K is w(K) = 2:r=o W(Ci)'

Note that this definition also holds for signed hypergroups and that W(Ci) 2: 1. Clearly Co has weight 1; the set of elements of weight 1 forms a group.

Two important examples of commutative hypergroups are associated with any finite group G, the class hypergroup K(G) consisting of probability measures on the conjugacy classes under convolution and the character hypergroup K( GA) consisting of normalized characters of Gunder pointwise multiplication. Here normalized means x( e) = 1.

For a finite set X = {Xl, ... ,xd let sX denote the collection of all formal sums of the form O:lXl + .. '+O:kXk where O:i 2: ° and 0:1 + .. '+O:k = 1. We assume no relations among the Xi; that is two sums O:lXl + .. '+O:kXk

and ßlXl + ... + ßkXk are equal if and only if O:i = ßi, i = 1, ... ,k. The set sX is an abstract simplex with vertices X.

For any X, y E sX and 0: E [0,1], the affine combination O:X+ (1- o:)y is a well-defined element of sX and more generally for any Xl, ... ,Xn E sX and 0:1, ... , O:n 2: 0, 0:1 + .. '+O:n = 1, the element O:lXl + .. '+O:nxn is a well-defined element of sx. Furthermore sX contains the distinguished point

1 1 C = -Xl + ... + -Xk

k k

148 v. S. Sunder and N. J. Wildberger

which we call the centroid of sX.

Definition 3. Let Af f(X) denote the set of all maps 'ljJ : sX -t sX that satisfy

'ljJ(ax + (1 - a)y) = a'ljJ(x) + (1 - a)'ljJ(y) V x, y E sX, a E [O,lJ.

Let Af f(X, c) be the set of all 'ljJ E Af f(X) that in addition satisfy

'ljJ(c)=c.

To any 'ljJ E Af f(X) we may associate the k x k nonnegative matrix T", = [tiil where

n

'ljJ(Xj) = LtijXj. i=1

Clearly the column sums of T", are 1, and if'ljJ E Af f(X, c) then in addition the row sums are also 1 . The map 'ljJ -t T", is thus a bijection from Af f(X) to the set of column stochastic k x k matrices which restricts to a bijection from Af f(X, c) to the set of all doubly stochastic k x k matrices. The latter, which we denote by Ok, is closed under affine combinations, multiplication, and adjoints T -t T*. It follows that we may define 'ljJ* for any 'ljJ E Af f(X, c) by

T",. = (T",)*

so that Af f(X, c) is also closed under affine combinations, multiplication, and adjoints.

Let IC C A be a finite hypergroup. Then sK may be regarded explicitly as the convex hull of IC in A and is closed under convex linear combinations, multiplication, and adjoints (given by *). We come now to the main definition.

Definition 4. An action of a hypergroup IC = {Co, cl, ... ,en} on a finite set X is a homomorphism 7r : sK -t Af f(X), that is, an assignment to every Ci E IC of an element 7r( Ci) E Af f(X) such that

( i) 7r( co) is the identity (ii) 7r(Ci)7r(Cj) = Lk nfj 7r(ck) where nfj are the structure constants

of IC. If in addition the image of 7r is in Af f(X, c) and

Actions of Finite Hypergroups and Examples

(iii) 7r(cr) = 7r(Ci)* for all Ci E K then we say that 7r is a *-action.

149

Definition 5. The character of an action 7r K ~ Af f(X) is the function

X(Ci) = tr 7r(Ci).

which we also write as the vector X = (X(eo), ... , X(cn )).

Note the useful fact that the character must always have non-negative entries.

Definition 6. An action 7r of K on X is reducible if there exists a nonempty subset Y C X, Y -=F X such that 7r(Ci)(Y) ~ sY VCi E K. Otherwise 7r is irreducible.

Definition 7. Two actions 7rl and 7r2 of K on sets Xl and X 2 respectively are equivalent if there exists a bijection 'IjJ : sXl ~ sX2 such that 'IjJ 0

7rl (Ci) = 7r2( Ci) o'IjJ for all Ci E K.

Recall that for any hypergroup K = {co, Cl, ... ,Cn } the element

1 n

eo = w(K) ~ W(Ci)Ci

of sK, called Haar-measure, satisfies CieO = eo Vi. The following is a restatement into this context of a result of [11].

Theorem 1. Suppose 7r is an action of a hypergroup K on a finite set X. Then the following are equivalent:

(i) 7r is irreducible (ii) T7r(eo) is a strictly positive matrix (iii) T7r(eo) is a rank one projection.

Let Jk denote the k x k matrix consisting of all 1 's and let h denote the k x k identity matrix.

Corollary 2. 1f 7r is an irreducible *-action of a hypergroup K of dimension k then T7r (eo) = tJk'

Proof. By Theorem 1 and the fact that 7r is a *-action, T7r (eo) is a rank one k x k doubly stochastic matrix. But there is only one such matrix .

•

150 v. S. Sunder and N. 1. Wildberger

3. Some general classes of examples

Example 1. Any hypergroup /C acts on itself by left multiplication. This is a *-action if and only if /C is a group.

Example 2. For any hypergroup /C = {co, Cl, ... ,Cn } and any subhypergroup .c, we may form the quotient hypergroup /C/.c. Then /C acts on the set /C/.c.

We provide some details. A subhypergroup .c of /C is a sub set of /C that contains the identity Co and is closed und er * and multiplication. The latter condition may be conveniently restated as folIows. For Ci, Cj, Ck E /C let us write Ck E CiCj iff n7,j > O. Then.c is closed under multiplication iff for any Li, lj E .c, Ck E l;lj implies Ck E .c.

Following [6], for Ci E /C the set

ci.c = {Ck ICk E Cilj for some lj E .c}

is called a left coset of.c. Left cosets are identical or disjoint. The reason is as folIows. If Cj E ci.c then Cj E Cilk for some lk E .c so that Co E CilkC;. Taking the involution * of both sides, we see that Co E c;fißi and thus Ci E cj.c.

The set of allieft cosets we denote by /C/.c. This set is a hypergroup in a natural way by defining

n

(Ci.c) (cj.c) L n~j(ck.c) k=O

where the latter sum can be rewritten as an affine combination of distinct cosets by collecting common terms. There is then a natural hypergroup homomorphism p : /C --t /C/.c defined by P(Ci) = ci.c. From Example 1 above we see that /C/.c acts on itself; combining this with P shows that /C also acts on /C/.c. More detail on the relation between /C and /C/.c may be found in ([14]).

The actions of /C obtained this way have a particular property which we abstract as folIows.

Definition 10. The action 7r of /C on a set X is strongly transitive if there exists an x E X such that X = 7r(/C) x.

Proposition 3. Given a hypergroup /C and a subhypergroup .c, the action of /C on /C/.c is strongly transitive. Conversely any strongly transitive action of /C arises this way from some subhypergroup .c c /C.

Actions of Finite Hypergroups and Examples 151

Proof. The first statement is obvious; just take x to be the coset L. Suppose then that 7r : K -+ Af f(X) is a strongly transitive action on a set X with distinguished element x as in Definition 10. Let L = {Ci E KI7r(Ci) x = x}. This is a subhypergroup of K. Consider the affine map '(jJ : s(Kj L) -+ sX given on the vertices by '(jJ(CiL) = 7r(Ci) x. This map is weH defined and estabishes a bijection between the two sets which commutes with the actions of K. •

Example 3. Let G be a finite group with conjugacy dasses Co {e}, Cl,"" C n and let K(G) = {co, Cl,"" Cn } be the dass hypergroup; recaH that this means that Ci is the element 16i l LgECi 9 in the group

algebra and that ci is 16i l LgECi g-l. Suppose that G acts on a finite set X = {X1, ... ,Xk}' Define 7r(Ci) E Aff(X,c) by

1 7r(Ci)(Xj) = IGI L g. Xj.

, gECi

Proposition 4. 7r is a *-action of K( G) on X. It is irreducible if and only if the action of G on X is transitive.

Proof. Clearly 7r(eo) is the identity.

so 7r is a homomorphism. The entry 7r(Ci)uv (identifying the operator with its matrix with respect to the ordered basis {Xl,"" X n }) is the coefficient of X u in the expression I~il LgECi 9 . XV' But this is the same

as the coefficient of Xv in the sum I~il L9ECi g-l xu which is 7r(c;)vu. Thus

so 7r is a *-action.

152 v. S. Sunder and N. J. Wildberger

If Y c X is a non trivial Gorbit then 1T(Ci)Y C sY for all i so that if the *-action 1T is irreducible then G acts transitively. Conversely if the *-action of G is transitive then 1T( eo) has strictly positive entries, so by Theorem 1, 1T is irreducible. •

4. The Basic Inequality

In this section we will prove the following result and some related facts.

Theorem 5. Suppose a commutative hypergroup K acts irreducibly on a finite set X. Then

lXI ~ w(K).

We will use some basic facts about harmonie analysis on a eommuta-tive hypergroup K = {co, Cl, ... ,Cn } CA from [13] whieh we now review. There exists a basis {eo, el, ... ,en } of A consisting of mutually orthog-onal idempotents (this me ans eiej = bi,jei) such that

Ciej = Xj(ci)ej for all Ci E K

for some complex-valued functions Xj. These functions are exactly the characters of K, that is, those functions X sueh that

(i) X(Ci)x(Cj) = LkntX(ck) (ii) X(c;) = X(Ci).

The set {Xo, Xl,.'" Xn} of eharacters of K is denoted by K II . Charaeters are orthogonal with respect to the inner product

1 n __

(I, g) = w(K) ~ W(Ci)!(Ci)g(Ci)

and form a signed hypergroup under pointwise multiplieation and complex conjugation.

The weights w(Xj) are still well-defined and positive, and w(KII ) = Lj w(Xj) = w(K). Furthermore we have explieit formulae for the relationships between the bases {co, ... ,cn } and {eo, ... ,en } of A :

In particular Xo == 1 and eo is as given previously. Assurne we now have a *-action of the commutative hypergroup K c

A on a set X = {Xl, ... , xd. Let B denote the complex vector space with basis X; that is B = {L7=1 ZiXi I Zi E Cl. The *-action n of K on X extends linearly to a *-action of A on the vector space B. Define Bj = n(ej)B and let dirn Bj = bj .

Definition 14. The vector b = [bo, ... , bnl will be called the multiplicity vector of the action.

Since the ej are orthogonal idempotents and Co = Lj ej, we get

B = eiJ=o Bj

where n( ej) is the projection onto Bj . Then

All the quantities involved in the right hand side of this expression are positive, except for Xj(Ci), which is a complex number of modulus at most one.

Thus we have

W(Xj) ~ bj < w(K) ~ w(e;) tr n(ci)

w(Xj) tr n(eo)

w(Xj)

since by assumption n is irreducible and so n( eo) is a rank one projection and has trace 1. Then

n n

lXI = l: bj ::; l: w(Xj) = w(Kf\) = w(K). j=O j=O

This proves our theorem and in addition establishes the following.

Proposition 6. (a) If bj = dirn Bj = dirn n(ej)B then

bj ::; w(Xj)·

(b) The character 01 the action is given by

n

X(Ci) = l: bjXj(Ci). j=O

154 V. S. Sunder and N. J. Wildberger

Definition 16. An action 1C' of a hypergroup K on a set X is maximal if lXI = w(K).

For a maximal action to exist, w(K) must be an integer. Maximal *-actions, if they exist, are of particular interest due to a connection with association schemes. We define these combinatorial objects here; for more information see [1], [3], [5J.

Definition 17. An association scheme on a finite set of k elements is a set {Ao, Al,"" An} of k X k matrices with the following properties:

(i) Each entry of any Ai is either ° or 1. (ii) Ao is the identity matrix. (iii) There exist non-negative integers p~ J' such that AiAj =

n I ' ~/=oPi,jAI'

(iv) The set {Ao, Al,"" An} is dosed under formation of adjoints (v) ~l=o AI = Jk .

If Ai< is the adjoint of Ai, then the number Wi = p?* i is called the valency of the 'i-th' dass of the association scheme. It is not hard to show that if we define e; = W;l Ai, then {co, Cl, ... ,en} forms a hypergroup of matrices.

Conversely, given a hypergroup K, when does it arise in this fashion from some association scheme? The answer is given, at least partially, by the following (for a proof see [12]).

Theorem 7. (a) Suppose that a hypergroup K admits a maximal *action 1C' : K -+ Aff(X). Then the matrices {A; = w(ci)T1r(c;),O :::;

i :::; n} define an association scheme, and K arises from that association scheme in the manner described above.

(b) Conversely if a hypergroup K arises from an association scheme in the manner described above, then K admits a maximal *-action.

5. Some specific examples

In the examples that follow, we will not distinguish between an affine map 1C'(e;) and the corresponding matrix T1r(Ci)'

Example 1. Hypergroups of order two

A hypergroup K = {co, cd with two elements is determined by the single equation

ci = O:Co + (1 - O:)Cl

where 0 < 0: ::; 1 is arbitrary. Then w(K) = (0: + 1)/0: and the Haar measure is

0: 1 eo = --co + --Cl·

0:+1 0:+1 Suppose that K *-acts irredueibly on a set X with lXI = k. By Corollary 9,

and so 0:+1

7f(cd = (0: + l)7f(eo) - O:7f(co) = -k-Jk - o:h.

Provided that Qtl ~ 0: (the inequality of Theorem 13 in this special ease), this is a doubly stoehastie matrix.

Note that

(0: + 1)2 J2 20:(0: + 1) J 21 k2 k - k k + 0: k

(0: + 1)(1 - 0:) J 21 k k+O: k

( (0:+1) ) o:h+(l-o:) k Jk-o:h

O:7f(co) + (1 - O:)7f(cd

so this is indeed a *-aetion. Summarizing, we have

Proposition 8. The hypergroup K = {co, Cl} with structure equation ci = O:Co + (1 - O:)Cl for 0: E (0,1] has exactly one irreducible *-action on any set X of size k ::; "'!l. This *-action is given by

0:+1 7f(Cl) = -k-Jk - o:h.

Example 2. K(83 )

The dass hypergroup of the symmetrie group 8 3 is K(83 )

{co, Cl, C2} with strueture equations


From Theorem 13 we know that K(S3) ean *-act irredueibly only a set X with lXI :S 6. From Theorem 12 we also know that there are irredueible *-aetions of dimensions k = 1,2,3 and 6. Are there any others?

First we note that C = {co, C2} is a subhypergroup of K of total weight 3, so that any irredueible *-aetion of K deeomposes upon rest riet ion to C into irreducibles of size 1,2 or 3.

Using Proposition 18, we find the irredueible 2 and 3 dimensional *-aetions of C are given by one of the following

Note that if lXI = k then

and is determined by 1f(C2)' This equation also shows that 1f I.c cannot eontain any 1 dimensional *-aetions unless k = 2, as otherwise a negative term would appear somewhere on the diagonal of 1f(Cl)'

If k = 2, there are 2 possibilities aeeording as 1f I.c is redueible or irredueible. These are:

or

1 [1 1] 1f(ct} ="2 1 1 1 [1 3] and 1f(C2) = 4 3 1 .

The first of these eorresponds to the *-action on G / H, where IHI = 3. If k = 3, 1f Ic must be irredueible so we get

1[111] 1f(ct} = - 1 1 1 3 111 which eorresponds to the *-aetion on G / H, where IHI = 2.

If k = 4, 1f le must split into 2 irreducibles of dimension 2, so again 1f(C2) is determined and we get

[

0011

1 10011

1f(CI) = 2 1 1 0 0

1 100

[1300] 1 3 1 0 0

and 1f(C2) = 4" 0 0 1 3 .

003 1

If k = 5, 1f le must split into irreducibles of dimension 2 and 3. But then 1f(C2) has two entries of ~ on the diagonal (thanks to the dimension 2 component) and the corresponding entries of 1f(CI) will be ~-~-~ x ~ < 0 which is impossible. Therefore no dimension 5 *-action exists.

If k = 6, 1f I.c must split into 2 pieces of dimension 3 by the above argument for k = 5. We obtain

0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0

1 0 0 0 1 1 1 1 1 1 0 0 0 0 1f(CI) = "3 1 1 1 0 0 0

and 1f(C2) = 2 0 0 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 1 1 0

This corresponds to the *-action of IC on G itself- a maximal *-action which corresponds to the association scheme given by the conjugacy classes of G. Summarizing, we have the following:

Proposition 20. Let IC = {co, Cl, C2} = IC(S3) with structure equations ci = ~co + ~C2' CIC2 = Cl and C~ = ~co + ~C2. Then IC has exactly the following irreducible *-actions:

(i) 1f(cd = 1f(C2) = [1]

(ii) 1f(CI) = [~ ~] and 1f(C2) = [~ ~]

(iii) 1f(CI) = ~ [i i] and 1f(C2) = ~ [~ ~]

(iv) *1) ~ I [: : n and ~(C2) ~ ~ [: ~ i 1

(v) *') ~ ~ [! ! ~ ~ j and ~(C2) ~ i [~ ~ ! ~ j

158 V.S. Sunder and N.l. Wildberger

0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0

(vi) 7r(CI) = ~ 0 0 0 1 1 1 and 7r(C2) = ~ 1 1 0 0 0 0

1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 1 1 0

We find that besides the 'expected' *-actions on the homogeneous spaces of G, we also have two others of dimensions 2 and 4.

Example 3. K.,(S~)

For OUf next example, we consider the character hypergroup of S3 given by K.,(S~) = {XO,XI,X2} with equations

Xo

X2 111 =lXo + =lXI + 2X2

Then W(XI) = 1 and W(X2) = 4 so that

112 eo = "6Xo + "6XI + 3"X2 .

.c = {Xo, Xl} is a subhypergroup of total weight 2 whose only nontrivial irreducible *-action is given by

The restriction of any *-action to .c is a sum of trivial and 2-dimensional irreducibles. Since from Corollary 9 we have

3 lk 1 1 7r(X2) = 2k - =lIk - 47r (XI)

we may check the possibilities for 7r(XI) which are consistent with the condition 7r(X2) E nk.

or

For k = 2, the possibilities are

7r(XI) = [~ ~] and 7r(X2) = ~ [~ i] 7r(XI) = [~ ~] and 7r(X2) = ~ [~ ~].

For k = 3 the possibilities are

[100] 1[0 1 i ] 1T(Xl) = 0 1 0 and 1T(X2) ="2 1 0 o 0 1 1 1

or

[1 0 0] 1 [ 0 2 n 1T(Xl) = 0 0 1 and 1T(X2) = - 2 1 010 4 2 1

For k ~ 4, 1T I C cannot contain any 1 dimensional *-actions since then by (*), a diagonal term of 1T(Xl) would be

3 1 1 - - - - - < O. 2k 4 4

Thus for k = 4 we get

[010 o 1 [ 1

1 3

~ 1 100 o 1 1 1 3

1T(Xl) = 0 0 0 ~ and 1T(X2) = 8 ~ 3 1 001 3 1

and for k = 5 there are no possibilities. For k = 6 we get

0 1 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1

1T(xd = 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 1 0 0 0 and 1T(X2) = 4" 1 1 0 0 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0

Summarizing, we have the following

Proposition 21. Let JC = {XO,Xl,X2} = JC(S;) with structure equa

tions X~ = Xo, XIX2 = X2 and X~ = ~XO + ~Xl + ~X2. Then JC has exactly the following irreducible *-actions:

(i) 1T(Xl) = 1T(X2) = [1]

(ii) 1T(Xl) = [~ ~] and 1T(X2) = ~ [~ ~]

(iii) 1T(Xl) = [~ ~] and 1T(X2) = ~ [~ ~]


o 1 0 0 0 0 1 0 0 000 000 100 o 0 1 000 o 0 0 0 0 1 o 0 0 0 1 0

Example 4. The Golden hypergroup

001 1 1 1 001 1 1 1 1 100 1 1 1 100 1 1 1 1 1 100 111 100

The Golden hypergroup arises from the association scheme of a pentagon viewed as a graph; by this we mean that the two non trivial classes are given by the relations of being neighbours or not being neighbours respectively. K = {co, Cl, C2} has structure equations

C2 1 1 -Co + -C2 I 2 2 1 1

CIC2 -Cl + -C2 2 2

c 2 1 1 -Co + -Cl 2 2 2

and character table

Co Cl C2

Xo 1 1 1

Xl 1 -1+V5 -1-V5 -4- -4-

X2 1 -1-V5 -1+V5 -4- -4-

Note that from the table we see that K/' ~ K and so that W(XI =

W(X2) = 2. We know from Theorem 13 that any irreducible *-action must be of dimension k ~ w(K) = 5.

If k = 2 there are two possibilities for the dimension vector; b = [1,1,0] or b = [1,0,1] which from the character table results in the following two possibilities for the character; X = (2, 3+4\1'5, 3-4\1'5) or

X = (2, 3-4\1'5, 3+4\1'5). Since a two dimensional doubly stochastic matrix is determined by its trace, we get the following possibilities:

[ 3+v's 5-v's] [3-v's 5+v's] 7r(C1) = 1/8 5 _ v's 3 + v's ,7r(C2) = 1/8 5 + v's 3 - v's '

[ 3-v's 5+v's] [3+v's 5-v's] 7r(C1) = 1/8 5 + v's 3 - v's ,7r(C2) = 1/8 5 - v's 3 + v's

Both of these can be easily checked to be *-actions. If k = 3 the situation is more subtle. We will show that no *-action 7r

exists by assuming the contrary. From the character table the only possibility for the multiplicity vector is b = [1,1,1]. Suppose that we place the simplex X on which K acts in a Euclidean plane as an equilateral tri angle whose vertices are on the circle of radius one about the origin O. Each of T = 7r(cd, U = 7r(C2) are then real symmetric linear transformations of this plane onto itself which commute and whose eigenvalues are known from the character table and the multiplicity vector. This means we may choose orthogonal coordinates (x, y) such that if rand s denote the quantities -11\1'5 and -1~\1'5 respectively then T and U are the linear transformations T(x, y) = (rx, sy) and U(x, y) = (sx, ry).

By assumption both T and U map the equilateral triangle into itself. In particular since the origin divides any median of this triangle in the ratio 2 : 1, for any vertex v = (cosO,sinO) we must have (Tv,v) ;:::: -~ and (Uv, v) ;:::: -~. Thus

r cos2 0 + s sin2 0 1

> 2

S cos2 0 + r sin2 0 1

> 2

The second of these equations may be manipulated easily to get

20< 1+V5 cos - 2v's

Now 1iJi ::; cos2 7r /6 so that I cos 01 ::; cos 7r /6 = -13/2. Similiarly

I sin 01 ::; -13/2. This means that 0 can only lie in the range 7r /6 ::; o ::; 7r /3 (mod 7r /2). But then one of 0 + 27r /3, 0 - 27r /3 lies outside

162 V. S. Sunder and N. J. Wildberger

this range mod 7T /2. But these are the other vertices of the equilateral triangle we are considering so the same argument must apply to them.

This contradiction shows that no irreducible *-action of dimension k = 3 exists.

If k = 4 the possibilities for the multi pli city vector b = [1,3,0] or b = [1, 2, 1] are easily dismissed; the first by Proposition 15 and the second by the fact that the corresponding character would be X = (4, 1+/5 , 1-4v'5). The same is true for the alternatives b = [1,0,3] and b = [1,1,2].

If k = 5 then the multiplicity vector is necessarily b = [1,2,2] and the associated character is X = [5,0,0]. By Theorem 7 we know that 27T(Cl) and 27T(C2) are 0, 1 symmetrie matrices which sum to J5 - h. This me ans that each must have exactly two l's in each row (and column). A bit of thought shows that up to isomorphism there is only one such arrangement and that it does indeed define a *-action.

0 1 0 0 1 1 0 1 1 0

1 1 0 1 0 0 1 0 1 0 1 1 7T(cd - 0 1 0 1 0 and 7T(C2) - 1 0 1 0 1

2 0 0 1 0 1 2 1 1 0 1 0 1 0 0 1 0 0 1 1 0 1

This is the association scheme associated to the graph of a pentagon.

References

[1] E. Bannai and T. Ito, Algebraie Combinatories I-Assoeiation sehemes, Benjamin and Cummings, Menlo Park, 1984.

[2] W. R. BIoom and H. Heyer, Harmonie Analysis of Probability Measures on Hypergroups, De Gruyter, Berlin, (1995).

[3] A. E. Brouwer, A. M. Cohen and A. Neumaier, Distanee- Regular Graphs, Ergeb. der Math., no 3. Band 18, Springer- Verlag, Berlin, 1989.

[4] C. F. Dunkl, The measure algebra of a locally eompaet hypergroup, Trans. Amer. Math. Soc., 179 (1973), 331-348.

[5] C. D. Godsil, Algebraie Combinatories, Chapman and Hall, London, 1993.

[6] R. I. Jewett, Spaces with an abstract convolution of measures Adv. Math., 18 (1975), 1-101.


[7] G. L. Litvinov, Hypergroups and hypergroup algebras,J. Soviet. Math. 38 (1987),1734-1761.

[8] K. A. Ross, Hypergroups and centers of measure algebras, Symposia Math. 22 (1977), 189-203.

[9] R. Spector, Mesures invariantes sur les hypergroupes, Trans. Amer. Math. Soc., 239 (1978), 147-165.

[10] V. S. Sunder, On the relation between subfactors and hypergroups, Applications of hypergroups and related measure algebras, Conternp. Math., 183 (1995),331-340.

[11] V. S. Sunder and A. K. Vijayarajan, On the non-occurrence of the Coxeter graphs ß2n+l, D2n+ 1 and E 7 as the principal graph of an inclusion of II1 factors, preprint.

[12] V. S. Sunder and N. J. Wildberger, On discrete hypergroups and their actions on sets, preprint (1996).

[13] N. J. Wildberger, Duality and Entropy for finite commutative hypergroups and fusion rule algebras, to appear.

[14] N. J. Wildberger, Lagrange 's theorem and integrality for finite commutative hypergroups with applications to strongly-regular graphs, to appear, J. of Alg.

Institute of Mathematical Sciences, Madras 600113, INDIA

School of Mathematics, University of New South Wales, Sydney, 2052, AUSTRALIA

Püsitivity of Tunin Determinants für

Orthogonal Polynomials

Ryszard Szwarc*

Abstract

The orthogonal polynomials Pn satisfy Thnin's inequality if P; (x) - Pn-l (x )Pn+l (x) ::::: 0 for n ::::: 1 and for all x in the interval of orthogonality. We give general criteria far orthogonal polynomials to satisfy Thnin's inequality. This yields the known results for classical orthogonal polynomials as well as new results, for example, for the q-ultraspherical polynomials.

1. Introduction

In the 1940s, while studying the zeros of Legendre polynomials Pn(x), Tunin [T] discovered that

(1)

with equality only for x = ±1. Szegö [Szl] gave four different proofs of (1). Shortly after that, analogous results were obtained for other classical orthogonal polynomials such as ultraspherical polynomials [Sk, S], Laguerre and Hermite polynomials [MN], and Bessel functions [Sk, S].

In [KS] Karlin and Szegö raised the quest ion of determining the range of parameters (a, ß) for which (1) holds for Jacobi polynomials of order (a, ß); i.e. denoting R~Ct,ß)(x) = P~Ct,ß)(x)/ P~Ct,ß)(l),

[R~Ct,ß)(xW - R~"'..:f)(x)R~~f)(x) :2 0, (2)

In 1962 Szegö [Sz2] proved (2) for ß :2 lai, a > -1. In aseries of two papers [GI, G2] Gasper extended Szegö's result by showing that (2) holds if and only if ß :2 a > -1.

*This work has been partially supported by KBN (Poland) under grant 2 P03A 03009.

1991 Mathematics Subject Classijication. Primary 42C05, 47B39 Key words and phrases: orthogonal polynomials, Tunin's inequality, recurrence

formula.

166 R. Szwarc

More reeently, attention has also turned to the q-analogues of the classical polynomials [BI1].

All the results mentioned above were proved using differential equations, that the classieal orthogonal polynomials satisfy. Therefore the methods eannot be used to extend (1) to more general orthogonal polynomials. In 1970 Askey [A, Thm. 3] gave a general eriterion for monie symmetrie orthogonal polynomials to satisfy the Tunin type inequality on the entire real line. His result, however, does not imply (1) for the Legendre polynomials beeause the latter are not monie in the standard normalization, and they do not satisfy Askey's assumptions in the monie normalization. In this paper we give general eriteria for orthogonal polynomials implying (1) holds for x in the support of eorresponding orthogonality measure. The assumptions are stated in terms of the eoeffieients of the reeurrenee relation that the orthogonal polynomials satisfy. They admit a very simple form in the ease of symmetrie orthogonal polynomials; i.e. the ease Pn( -x) = (-l)npn (x). In partieular, the results apply to all the ultraspherieal polynomials, giving yet another proof of Tunin's inequality for the Legendre polynomials.

It turns out that the way we normalize the polynomials is essential for the Tunin inequality to hold. The results eoneerning the classical orthogonal polynomials used the normalization at one end point of the interval of orthogonality, e.g. at x = 1 for the Jaeobi polynomials and at x = 0 for the Laguerre polynomials. We will also use this normalization and will show that this ehoiee is optimal (Proposition 1). However, the reeurrenee relation for the polynomials normalized in this way may not be available explieitly. This is the ease of the q-ultraspherieal polynomials. We give a way of overeoming this obstacle (Corollary 1). In partieular, we prove the Tunin inequality for all q-ultraspherieal polynomials with q > O. These polynomials have been studied by Bustoz and Ismail [BIl] but with a normalization other than at x = 1. The same method is applied to the symmetrie Pollaezek polynomials, studied in [BI2], again with different normalization.

In Seetion 6 we prove results for nonsymmetrie orthogonal polynomials (Thm. 4). The ass um pt ions again are given in terms of the eoeffieients in a three term reeurrenee relation but they are much more involved.

In Seetion 7 we state results eoneerning polynomials orthogonal on the positive halfaxis. In partieular they ean be applied to the Laguerre polynomials of any order 0:.

Positivity of Turan Determinants for Orthogonal Polynomials 167

2. Basic formulas

Let Pn be polynomials orthogonal with respect to a probability measure on IR. The expressions

ßn(x) = p~(x) - Pn-l (x)Pn+l (x) n = 0, 1, ... , (3)

are called the Tunin determinants. Our goal is to give conditions implying the nonnegativity of ßn(x) for x in the support of the orthogonality measure.

The first problem we encounter is that the orthogonality determines the polynomials Pn up to a nonzero multiple. The sign of ßn(x) may change if we multiply each Pn by different nonzero constants. We will normalize the polynomials Pn to obtain the sharpest results possible. N amely, we will assume that

Pn(a) = 1

at a point a in the support of the orthogonality measure. In this way the Tunin determinant vanishes at x = a.

Our main interest is focused on the case when the orthogonality measure is supported in a bounded interval. By an affine change of variables we can assume that this interval is [-1, 1]. In that case we set a = 1. Since the polynomials Pn do not change sign in the interval [1, +00) they have positive leading coefficients.

Assume that the polynomials Pn are orthogonal, with positive leading coefficients and Pn (1) = 1. Then they satisfy the three term recurrence relation

xPn(x) = I'nPn+l(X) + ßnPn(X) + OnPn-l(X) n = 0, 1, ... , (4)

with initial conditions P-l = 0, Po = 1, where On, ßn, and l'n are given sequences of real valued coefficients such that

00 = 0, 0n+l > 0, l'n > ° for n = 0, 1, ....

Plugging x = 1 into (4) gives

On + ßn + l'n = 1 n = 0,1, .... (5)

Proposition 1. Let the polynomials Pn satisfy (4) and (5). Then

I'nßn = I'nP~ + OnP~_l - (x - ßn)Pn-lPn, (6) I'nßn = (Pn-l - Pn)[(')'n-l -l'n)Pn + (on - On-l)Pn-l] + On-l ß n-l,(7)

I'nßn = (Pn - Pn-l)(l'nPn - OnPn-l) + (1 - X)Pn-lPn, (8)

for n = 1, 2, .. , .

168

Proof. By (4) we get

'"'Inßn '"'InP;' - '"'InPn-1 [(x - ßn) - anPn-l]

'"'InP;' + anP;'_1 - (x - ßn)Pn-IPn

R. Szwarc

'"'InP;' + anP;'_1 - (ßn-l - ßn)Pn-IPn - (x - ßn-l)Pn-IPn.

N ow applying (4), with n replaeed by n - 1, to the last term yields

The use of ßn-l - ßn = bn - '"'In-I) + (an - an-I)

eoncludes the proof of (7). In order to get (8) replaee ßn with 1- an - '"'In in (6). •

3. Symmetrie polynomials

We will eonsider first the symmetrie orthogonal polynomials, i.e. the orthogonal polynomials satisfying

(9)

Theorem 1. Let the polynomials Pn satisfy

n = 0,1, .... (10)

with P-I = 0, Po = 1, where ao = 0, an+! > 0, '"'In > 0, and

an + '"'In = a n = 0,1, ....

Assume that either (i) or (ii) is satisfied where

a (i) an is nondecreasing and an ::; 2" for n = 1, 2, ....

a (ii) an is nonincreasing and an ~ 2" for n = 1, 2, ....

Then for - a ::; x ::; a, n = 0, 1, ... ,

and the equality holds if and only if n ~ 1 and x = ±a. Moreover if (i) is satisfied then

for lxi> a, n = 1, 2, ....

Positivity of Tunin Determinants for Orthogonal Polynomials 169

Proof. By changing variable x -t ax we can restrict ourselves to the case a = L We prove part (i) only, because the proof of (ii) can be obtained from that of (i) by obvious modifications.

By assumption we have Pn(l) = 1 and Pn(-l) = (_l)n. Hence ß n (± 1) = O. Assurne now that lxi< L By (9) it suffices to consider 0 ::; x < L The proof will go by induction. We have rlßl(X) =

l"tl(l-r) 2: O. Now assume ßn-1(x) > O. In view of ßn = 0 and l"tn - l"tn-l = rn-l - rn, Proposition 1 implies

2 2 rnPn + l"tnPn_l - XPn-lPn,

(l"tn - l"tn-l)(P~_l - p~) + l"tn-lßn-l'

(11)

(12)

By (11) and the positivity of x we may restrict ourselves to the case Pn-l(X)Pn(x) > O. We will assume that Pn-l(X) > 0 and Pn(X) > 0 (the case Pn-l(X) < 0 and Pn(x) < 0 can be dealt with similarly). By (12) and by the induction hypothesis it suffices to consider the case Pn-l(X) < Pn(x), since by assumption (i) we have an-I::; an. In that case since rn = 1 - an 2: ~ 2: an we get

Now we apply (8) and obtain

The proof of part (i) is thus complete. We turn to the last part of the statement. Let (i) be satisfied and

lxi> 1. By symmetry we can assume x > 1. As before we proceed by induction. We have

Assume now that ß m (x) < 0 for 1 ::; m ::; n - 1. Since Pn (1) = 1 and the leading coefficients of Pn's are positive, the polynomials Pn are positive for x > 1. Thus

for 1 ::; m ::; n - 1. Hence

Pm(X)

Pm-l (x)

Pn(X) > Pl(X) _ () _ ... 2: ()-x>1.

Pn-l X Po x

170 R. Szwarc

Now by (12) we get

• Remark. The second part of Theorem 1 is not true under assumption (ii). Indeed, by (10), the leading coefficient of the Tunin determinant /'nßn(x) is equal to /'1"2 .. '/';;~l(etn-l - etn). Thus ßn(X) is positive at infinity for n 2:: 2. One might expect that in this case ßn(x) is nonnegative on the whole real axis, but this is not true either. Indeed, it can be computed that

One can verify that under assumption (ii) we have

2

r:= et l /'2 > 1. /'2 - /'1

(Actually r 2:: 1 follows from Theorem l(ii).) Hence ß2(X) < ° far 1< x < r.

Sometimes we have to deal with polynomials which are orthogonal in the interval [-1,1] and normalized at x = 1, but the three term recurrence relation is not available in explicit form. In such cases the following will be useful.

Corollary 1 Let the polynomials Pn satisfy

n = 0,1, ... ,

with P-l = 0, Po = 1 and eto = 0. Assume that the sequences etn and etn + /'n are nondecreasing and

. 1 hm etn = -2a

n--+oo 1. 1 -1 1m /'n = -a ,

n--+oo 2

where 0 < a < 1. Then the orthogonality measure for Pn is supported in the interval [-1,1].

Assume that in addition at least one of the following holds

(i) /'n is nondecreasing,

(ii) /'0 2:: 1.

Then

ßn(X) = p~(x) - Pn-l(X)Pn+l(X) 2:: 0 <===} -1::; x ::; 1,

where Pn(x) = Pn(x)/Pn(1).

Prüüf. First we will show that Pn (1) > O. In view of symmetry of the polynomials this will imply that the support of the orthogonality measure is contained in [-1, 1J.

We will show by induction that Pn(I)/Pn-l(l) 2: a > O. We have

Po(l) _ < ~( + -1) < -1 PI (1) - 10 - 2 a a - a .

Assume that Pn(1)/Pn-l(l) 2: a. Then from the recurrence relation we get

Pn+l(l) = ~ (1- anPn- 1(1)) > ~ (1 - a-Ian). Pn(l) In Pn(l) - In

On the other hand

In 1 -1

(an + In) - an ::; "2(a + a ) - an

1 < "2(a + a- I) - a-2an + (a- 2 - l)an

1 1 < "2(a + a- I) - a-2an + (a- 2 - l)"2a

a- I (1- a-Ian).

Therefore Pn+1(I) > a. Pn(1) -

Now we show that Cn = p~(1) - Pn-I(I)Pn+1(I) > 0 by induction. Assume (i). Similarly to the proof of Proposition 1 we obtain

This implies Cn > 0 for every n. Assume now that (ii) holds and that Cm > 0 for m ::; n - 1. Hence

the sequence Pm+l(I)/Pm(l) is positive and nonincreasing for m ::; n-1. In particular Pn (1) / Pn-I (1) ::; PI (1) / Po (1) = 10 I ::; 1. Therefore Pn (1) ::; Pn-I(I). Rewrite (13) in the form

InCn = [(an + In) - (an-l + In-dlp~(I) +(an - an-I)[P~_I(I) - p~(1)l + an-ICn-l·

Thus Cn > O. We have shown that, in both cases (i) and (ii), we have Cn > 0 and

hence the sequence Pn-I(I)/Pn(1) is nondecreasing. Denote its limit by

172 R. Szwarc

r. Now plugging x = 1 into the recurrence relation for Pn, dividing both sides by p .. (I) and taking the limits gives

1 1 = 2" [ar + (ar)-l].

Thus r = a-1 . Let _ ( ) _ Pn(x) Pn X - Pn(1)"

From the recurrence relation for Pn we obtain

(14)

where _ Pn-l(l) an = Pn(1) .an,

_ Pn+l(l) ''ln = Pn(l) ,no

By plugging x = 1 into (14) we get

Since both Pn-l(I)/Pn(l) and an are nondecreasing, so is Ein. Moreover it tends to 1/2 at infinity because the first of its factors tends to a-1

while the second tends to a/2. Therefore the polynomials Pn satisfy the assumptions of Theorem l(i). This completes the proof. •

4. The best normalization

Assurne that the polynomials Pn satisfy (4) and (5). By multiplying each Pn by a positive constant an we obtain polynomials p~O"n)(x) = anPn(x). The positivity of Turan's determinant for the polynomials Pn is not equivalent to that for the polynomials p~O"n). However, it is possible that the positivity of Turan's determinants in one normalization implies the positivity in other normalizations. It turns out that the normalization at the right most end of the interval of orthogonality has this feature.

Proposition 2. Let the polynomials Pn satisfy (4) and (5). Assume that

p~(x) - Pn-l(X)Pn+l(X) 20, -1 ~ x ~ 1, n 2 1.

Let p~O")(x) = anPn(x), where an isa sequence of positive constants. Then

{p~)(X)}2 - P:~l (x)p~11 (x) 20, -1 ~ x ~ 1, n 2 1

if and only if

Positivity 0/ Turan Determinants for Orthogonal Polynomials 173

Proof. We have {p~)(X)}2 - P~~1(X)P~'11(X)

= (11~ - l1n-10"n+l)P~(X) + O"n-lO"n+l(P~(X) - Pn-l(X)Pn+l(X».

This shows the "ii" part. On the other hand, sinee (3) is equivaIent to Pn(l) = 1 for n 2 0, we obtain

{p~<Tn)(1)}2 - p~{(l)p~+{(l) = ~ - O"n-lO"n+1.

This shows the '"only Ir' part. •

Rernark. Proposition 2 says that if the Turan inequality holds for the polynomials normalized at x = 1 then it remains true for any other normalization if and only if it holds only at the point x = 1, b~'Ca;use p~"")(l) = O'n.

5. Applications to special symmetrie polynomials

We will test Theorem 1 on three classes of polynomials: ultraspherieal, q - ultraspherieal and symmetrie PoIlaezek polynomials.

The positivity of Tunin's determinants for the first ease is weIl known (see [E, p. 209]). The ultraspherieal polynomials C~A) are orthogonal in the interval (-1,1) with respeet to the measure (1 - x 2)A - (1/2) dx, where A > -~. When normalized at x = 1 they satisfy the reeurrenee relation

etA) _ n + 2A etA) n etA) x n - 2n+2A n+l + 2n+2A n-1·

It ean be eheeked easily that Theorem l(i) or (ii) applies aeeording to A 2 0 or A :S O.

Let us turn to the q-ultraspherieal polynomials. They have been studied by Bustoz and Ismail [BIl] but with a normalization other than the one at the right end of the interval of orthogonality. We will exhibit that our normalization Is sbarper in the sense that we ean derive the results of [BIl] from ours. Moreover, we will have no restrietions on the parameters other than tha:t q be positive.

In standard normalization the q~ultraspherieal polynomials are denoted by Cn(x; ßlq) and they satisfy the recurrenee relation

1 - qn+1 1 _ ß2qn-1 2xCn(x;ßlq) = 1 ß Cn+l(x;ßlq) + 1 ß Cn- 1(x;ßlq)· (15) _ qn _ qn

The orthogonality measure is known explieitly (see [AI], [AW, Thm. 2.2 and Sect. 4] Of [GR, Seet. 7.4]). When IßI, Iql < 1 it is absolutely eontinuous with respeet to the Lebesgue measure on the interval [-1,1].

174 R. Szwarc

Theorem 2. Let 0 < q < 1 and IßI < 1. Let i\(x; ßlq) denote the q - ultraspherical polynomials normalized at x = 1, i. e.

Let

Then Ön(x; ßlq) 2: 0 if and only if - 1 ::; x ::; 1,

with equality only for x = ±1.

Proof. The main obstacle in applying Theorem 1 lies in the fact that the values Gn (1; ßlq) are not given explicitly. Therefore, we cannot give explicitly the recurrence relation for Cn(x; ßlq).

We will break the proof into two subcases.

(i) 0< ß < 1. Introduce the polynomials

Then by (15) we obtain

Observe that 1

an + 'Yn = _(ßl/2 + ß-l/2). 2

Moreover an is nondecreasing and converges to ~ßl/2. Finally

Therefore we can apply Theorem l(ii) with a = ßl/2.

(ii) -1 < ß ::; O. Introduce the polynomials

Positivity oJ Turan Determinants Jor Orthogonal Polynomials 175

Then by (15) we obtain

xPn = I'nPn+l + anPn-l,

1- ß2qn

an = 2(1 _ ßqn) 1- qn

I'n = 2(1 - ßqn)"

Sinee both an and I'n are inereasing sequenees eonvergent to 1 we ean apply Corollary l(i) with a = 1. •

We turn now to the symmetrie Pollaezek polynomials P;(x; a). They are orthogonal in the interval [-1, 1] and satisfy the reeurrenee relation

A n + 1" n + 2A - 1 " xPn (x; a) = 2(n + A + a) Pn+1 (x; a) + 2(n + A + a) Pn- 1 (x; a),

where the parameters satisfy a > 0, A > 0. We eannot eompute the value P;(l; a) in order to pass direetly to normalization at x = 1. Instead, we eonsider another auxiliary normalization. Let

n! " Pn(x) = (2A)n Pn (x; a),

where (P)n = p(p + 1) ... (p + n - 1). Then the polynomials Pn satisfy the reeurrenee relation

n+2A n XPn = 2(n + A + a)Pn+l + 2(n + A + aln- 1 '

Observe that the assumptions of Corollary 1 (i) or (ii) are fulfilled aeeording to A ;::: a or A ~ a. Therefore we have the following.

Theorem 3. Let A > 0, a > 0. Let P;(x; a) denote the Pollaczek polynomials normalized at x = 1, i.e.

Then

-" 2 -" -" {Pn(x;a)} - Pn_1(x;a)Pn+1(x;a);::: 0 iJ and only iJ -1 ~ x ~ 1,

with equality only Jor x = ± 1.

6. Nonsymmetrie polynomials orthogonal in [-1,1]

In this seetion we assume that polynomials Pn satisfy (4) and (5) with ßn not neeessarily equal to 0 for all n.

176 R. Szware

Theorem 4. Let polynomials Pn satisJy (4) and (5). Let

(16)

Assume that Jor eaeh n ~ 2 one oJ the Jollowing Jour eonditions is satisfied.

(i)

or

(ii)

or

(iii)

(iv)

Then

an-I ::; an ::; rn ::; rn-I,

ßn + 1 + J(ßn + 1)2 - 4anrn < an - an-I 2rn rn-I - rn

(ßn + 1)2 - 4anrn < o.

an-I:::: an :::: rn ~ rn-I,

ßn + 1 - J(ßn + 1)2 - 4anrn > an-I - an 2rn rn -rn-I

(ßn + 1)2 - 4anrn < o.

1 rn-I ~ rn ~ 2'

an - an-I < an < 1 rn - rn-I - rn -

an - an-I an 1 or > - > . rn - rn-I - rn -

or

Proof. The proof will go by induction. Combining (8) and (4) for n = 1 gives

'Yhl6.1(X) = (1 - x)[('")'o - 'Yd(x - ßo) + anoJ. Now using (5) gives that the positivity of 6.1 (x) in the interval [-1, 1 J is equivalent to (16).

Fix x in [-1, IJ and assume that 6.n (x) ~ O. Consider two quadratic functions

Set

A(t)

B(x; t) = (t + 1){('")'n - 'Yn-1)t - (an-1 - an)},

'Yne - (ßn - x)t + an·

t = _ Pn(x) . Pn-1 (x)

By Proposition 1 it suffices to show that for any t the values A(t) and as B(x; t) cannot be both negative. In order to achieve this we have to look at the roots of these functions. The roots of A(t) are -1 and (an-1 - an)/('")'n - 'Yn-d; hence they are independent ofx. The roots of B(x; t) have always the same sign and are equal to

r~l)(x) = ßn - x - J(ßn - X)2 - 4an'Yn

(17) 2'Yn

r~2)(x) = ßn - x + J(ßn - x)2 - 4an'Yn

(18) 2'Yn

Since the function u I-t u + Ju2 - a2 , a > 0, is decreasing for u :::; -a and increasing for u ~ a we have

r~1)(I) :::; r~l)(x) :::; r~2)(x) :::; r~2)(I) if ßn - x:::; 0, (19)

r~l)(-I) :::; r~l)(x) :::; r~2)(x) :::; r~2)(-I) if ßn - x ~ 0, (20)

provided that (ßn -x)2-4an'Yn ~ O. Thus B(x; t) < 0 implies B(I; t) < 0 (B ( -1; t) < 0 respectively) if ßn - x :::; 0 (ßn - x ~ 0 respectively). Hence it suffices to show that the values A(t) and B(I; t) (the values A(t) and B ( -1; t) respectively) cannot be both negative if ßn - x :::; 0 (ßn - x 2: 0 respectively). We will break the proof into two subcases.

(a) ßn - x :::; -2Jan'Yn. In view of (8) and (6 )the roots of B(I; t) are -1 and -~. Byanalyzing

'l'n

the positions of these numbers with respect to the roots of A(t) one can easily verify that under each of the four assumptions (i) through (iv) the

178

values A(t) and B(l; t) cannot be both negative.

(b) ßn - x::; -2..janrn.

R. Szwarc

We examine the signs of A(t) and B( -1; t). Consider (i), (ii) and (iii). By analysing the mutual position of the roots of B( -1; t) and A(t) one can verify that A(t) and B(-l;t) cannot be both negative.

In case (iv) we have that B( -1; t) ;::: 0 because

(1 + ßn)2 - 4anrn = (2 - an - rn)2 - 4anrn ::; o.

• Remark 1. The assumption (iv) in Theorem 4 does not imply that the support of the orthogonality measure corresponding to the polynomials Pn is contained in [-1,1]. By (5) we have Pn(1) = 1 which implies that the support is located to the left of 1. However, it can extend to the left side beyond -1.

Remark 2. If we assume that ßn = 0 for n ;::: 0, then Theorem 4 reduces to Theorem 1. Indeed, in this case we have

ßn + 1 - J(ßn + 1)2 - 4anrn

2rn . ( an) mm 1, rn '

ßn + 1 + )(ßn + 1)2 - 4anrn

2rn

Example. Set

1 1 an = 2 - n + 2'

1 1 rn = 2 + 2(n + 2)'

1 ßn = 2(n + 2)

We can check easily that condition (16) is satisfied. We will check that also the assumptions (iii) are satisfied for every n ;::: 2. Clearly we have an-I ::; an ::; "(n ::; rn-I· Moreover

ßn + 1 + )(1 + ßn? - 4anrn

2"(n ßn + 1 2 an - an-I < --< = .

rn - rn-I - rn

Let Pn(x) satisfy (4). By Theorem 4(iii)

p~(x) - Pn-I (X)Pn+l (x) ;::: 0 for - 1 ::; x ::; 1.

Positivity of TUT/in Determinants for Orthogonal Polynomials 179

Let us determine the interval of orthogonality. Since CYn + ßn + "In = 1 we have Pn(1) = 1. Thus the support of the corresponding orthogonality measure is located to the left of 1. Actually the support is contained in the interval [-1,1]. Indeed, it suffices to show that Cn = (-l)npn( -1) > 0. We will show that Cn 2:': Cn-l > ° by induction. We have Co = 1. Assume Cn 2:': Cn-l > 0. Then by (4)

(1 + ßn)cn - CYnCn-l 2:': (1 + ßn - cyn)cn

> (1 - ßn - cyn)cn = "InCn·

Thus Cn+l 2:': Cn > 0.

7. Polynomials orthogonal in the interval [0, +00).

Let Pn be polynomials orthogonal in the positive halfaxis normalized at x = 0, i.e. Pn(O) = 1. Then they satisfy the recurrence relation of the form

with initial conditions P-l = 0, Po = 1, where CYn, and "In are given sequences of real coefficients such that

CYo = 0, "10 = 1, CYn+! > 0, "In > 0, for n = 0, 1, .. . . (22)

Theorem 5. Let polynomials Pn satisfy (21) and (22), and let

Assume that one of the following two conditions is satisfied.

(i) CYn :::; "In

(ii) CYn 2:': "In

Then ~n(x) = p~(x) - Pn-l (X)Pn+l (x) 2:': ° for x 2:': 0.

Proof. Let qn(x) = Pn(1- x). Then by (21) we obtain

We have qn(1) = 1. Thus the assumptions (iv) of Theorem 4 are satisfied for every n. From the proof of Theorem 4(iv) it follows that

180 R. Szwarc

q~(X) - qn-l (X)qn+l (x) 2 0 for x ~ 1 (the assumption x 2 -1 is inessential). Taking into account the relation between Pn and qn gives the conclusion. •

A special case of Theorem 5 is when an - an-l = 'Yn - 'Yn-l for every n. In this case, applying (8) gives the following.

Proposition 3. Let polynomials Pn satisfy (21) and (22), and let

n21.

Then

2() () ()_~(ak-ak-r)akak+l ... an-l( () ())2 Pn X -Pn-l X Pn+l X - ~ Pk X -Pk-l X . k=l 'Yk 'Yk+l ... 'Yn

In particular, if an 2 an-l for n 2 1, then

p~(X) - Pn-l (X)Pn+l (x) 20 for - 00 < X < 00,

where equality holds only for X = o.

Example. Let Pn(x) = L~(x)/ L~(1), where L~(x) denote the Laguerre polyno

mials of order a > -1. Then the polynomials Pn satisfy

XPn = -(n + a + I)Pn+l + (2n + a + I)Pn - npn.

Then an - an-l = 'Yn - 'Yn-l = 1, n 2 1.

Thus Proposition 3 applies. The formula for P;' - Pn-lPn+l in this case is not new. It has been discovered by V. R. Thiruvenkatachar and T. S. Nanjundiah [TN] (see also [AC, 4.7]).

Acknowledgements. I am grateful to J. Bustoz and M. E. H. Ismail for kindly sending me a preprint of [BI2]. I thank George Gasper for pointing out the references [AC, TN].

References

[AC] Al-Salam W.; Carlitz, L.: General Tunin express ions for certain hypergeometric series, Portugal. Math. 16, 119-127 (1957).

Positivity of Tunin Determinants for Orthogonal Polynomials 181

[A] Askey, R: Linearization of the product of orthogonal polynomials, Problems in Analysis, R Gunning, ed., Princeton University Press, Princeton, N.J., 223-228 (1970).

[AI] Askey, R,Ismail, M.E.H.: A generalization of ultraspherical polynomials, in Studies in Pure Mathematics, P. Erdös, ed., Birkhäuser, Basel, 55-78 (1983).

[AW] Askey, R, Wilson, J.A.: Some basic hypergeometrie orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985).

[BIl] Bustoz, J., Ismail, M.E.H.: Turaan inequalities for ultraspherical and continuous q-ultraspherical polynomials, SIAM J. Math. Anal. 14,807-818 (1983).

[BI2] Bustoz, J., Ismail, M.E.H.: Turan inequalities for symmetrie orthogonal polynomials, preprint (1995).

[E] Erdelyi, A.: "Higher transcendental functions," vol. 2, New York, 1953.

[GI] Gasper, G.: On the extension of Tunin's inequality to Jacobi polynomials, Duke. Math. J. 38, 415-428 (1971).

[G2] Gasper, G.: An inequality of Turan type for Jacobi polynomials, Proc. Amer. Math. Soc. 32, 435-439 (1972).

[GR] Gasper, G., Rahman, M.: "Basic Hypergeometrie Series," Vol. 35, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 1990.

[KS] Karlin, S.; Szegö, G.: On certain determinants whose elements are orthogonal polynomials, J. d'Analyse Math. 8, 1-157 (1960/61).

[MN] Mukherjee, B.N; Nanjundiah, T.S.: On an inequality relating to Laguerre and Hermite polynomials, Math. Student. 19, 47-48 (1951).

[Sk] Skovgaard, H.: On inequalities of the Turan type, Math. Scand. 2, 65-73 (1954).

[S] Szasz, 0.: Identities and inequalities concerning orthogonal polynomials and Bessel functions, J. d'Analyse Math. 1, 116-134 (1951).

[Szl] Szegö, G.: On an inequality of P. Turan concerning Legendre polynomials, Bull. Amer. Math. Soc. 54, 401-405 (1948).

[Sz2] Szegö, G.: An inequality for Jacobi polynomials, Studies in Math. Anal. and Related Topics, Stanford Univ. Press, Stanford, Calif., 392-398 (1962).

182 R. Szwarc

[TN] Thiruvenkataehar, V.R.; Nanjundiah, T.S.: Inequalities eoneerning Bessel functions and orthogonal polynomials, Proe. Indian Aead. Sei. 33,373-384 (1951).

[T] Tunin, P.: On the zeros of the polynomials of Legendre, Casopis Pest. Mat. 75, 113-122 (1950).

Institute of Mathematies Polish Aeademy of Scienees ul. Kopernika 00-950 Wrodaw, Poland

Current address: Institute of Mathematies Wrodaw University pi. Grunwaldzki 2/4 50-384 Wrodaw, Poland

Wavelets on Hypergroups

K. Trimeche

Abstract

We consider hypergroups K satisfying certain conditions. Important examples of such hypergroups are the double coset hypergroup, the Chebli-Trimeche hypergroup and the hypergroup associated with spherical mean operator. We define on K wavelets and a continuous wavelet transform, we prove Plancherel and inversion formulas for this transform, and using coherent states we characterize the image space of this transform.

Introduction

Hypergroups are locally compact spaces with a group-like structure on which the bounded measures convolve in a similar way to that on a locally compact group. In fact, a hypergroup K can be viewed as a probabilistic group in the sense that to each pair x, y of points of K there exists a probability measure 8x * 8y on K with compact support, not necessarily equal to the Dirac measure 8xy , for a composition x, y in K, such that (x, y) ---t Supp (8x * 8y) is a continuous mapping from K x K into the space of compact subsets of K. In place of the natural left translation of a function j by x, available in the group case, we deal in a hypergroup with a generalized (left) translation defined by

(1) Tx(f)(y) = L j(z)d(8x * 8y )(z)

for all y in K. For commutative hypergroups K a substantial body of Harmonie

Analysis has been built, where the convolution product, Fourier transform and Fourier-Plancherel trans form are available as important technical tools (see [1], [8]). In this work we consider commutative hypergroups for which their Plancherel measure satisfies certain conditions. Important examples of such hypergroups are

184 K. Trimeche

• The double coset hypergroup K = K\ G / K where G is a noncompact connected real semisimple Lie group with finite center and K is a compact subgroup of G (see [1], [6], [7]).

• The space K = [0, +oo[ with operation inherited from a singular differential operator. Such a hypergroup is called a ChebliTrimeche hypergroup (see [1], [2], [18], [19], [20]).

• The space K = [0, +oo[ xIRn with operation inherited from the spherical mean operator (see [11], [15]).

The continuous wavelet transform for functions on IRn is an integral transform for which the kernel is the dilated translate of a socalled wavelet g, a quite arbitrary square integrable function on IRn . A Plancherel formula for this transform is obtained, and in the course of its derivation we naturally arrive at an admissibility assumption for the wavelet g. There follow a Parseval and an inversion formula. Wavelet transforms have widespread applications, ranging from signal analysis in geophysics and acoustics, to quantum theory and pure mathematics. (See [3], [9], [10].)

It is a natural quest ion to ask whether there exist eontinuous wavelet transforms on hypergroups.

U sing the generalized translation (1) and the dilation 0 btained by Fourier transform we show that it is possible to define continuous wavelet transforms on hypergroups. We prove for these transforms Planeherel and inversion formulas, and we discuss them in the framework of eoherent states.

We eonclude this introduetion with a summary of the eontent of this paper.

In the first section we recall some results about hypergroups and the Harmonie Analysis on them (convolution product, Fourier transform, Plancherel and inversion formulas, Planeherel theorem, ... ).

In the second section we give the three examples of hypergroups eited above.

In the third section we define wavelets and continuous wavelet transforms on hypergroups and we prove for these transforms Plancherel and inversion formulas, and using coherent states we characterize the image spaces of these transforms.

Wavelets on Hypergroups 185

1. Harmonie analysis on hypergroups

1.1. Definition of hypergroups

Let K be a locally compact topological space with an operation *, called convolution, on the space M(K) of bounded complex Radon measures on K, which is a bilinear and separately continuous mapping from M(K) x M(K) into M(K) which preserves probability measures (i.e. Ml(K) x Ml(K) c Ml(K)). (K, *) is a hypergroup in the sense of Jewett (see [8]) ifthe following properties are satisfied for all x, y, z E K:

Cl: Ox * (Oy * Oz) = (Ox * Oy) * Oz, where Ox is the Dirac measure at the point x.

C2 : There exists an element e E K such that

C3 : There exists a continuous involution x --t x- of K such that e E Supp (ox * Oy) if and only if x = y- and the canonical extension of this involution to the space M(K) satisfies

C 4: The mapping (x, y) --t Ox * Oy is weakly continuous from K x K into M(K).

C5 : The support of Ox * Oy is compact.

C6 : The mapping (x, y) --t Supp (Ox * Oy) from K x K into the set of non-empty compact subsets of K is continuous. The hypergroup (K, *) is commutative if

8x * 8y = 8y * 8x , for all x, y E K.

A particular case is when x = x-, and we say then that K is hermitian. In the following sections we consider only commutative hyper

groups K. A commutative hypergroup always has a Haar measure, i.e. a positive Radon measure m such that for all f continuous on K and with compact support we have

v xE K, (m,8x * j) = (m,j)

where 8x * f denotes the function defined by

(Ox * J)(y) = (Oy * Ox-, J).

186 K. Trimeche

The measure m is unique up to a multiplicative constant and Supp m = K (see [1], [8]).

1.2. Generalized translation operator and convolution on hypergroups

Notations: We denote by

(i) C(K) the space of continuous functions on K.

(ii) Cc(K) the subspace of C(K) consisting of compactly supported functions.

(iii) LP(K, m), p E [1, +00], the space of measurable functions on K such that

Ilfllp = ([ If(x)IPdm(x)) l/p < +00, 1 ~ p < +00,

Ilflloo = esssup If(x)1 < +00. xEK

Definition 1.1. Let f be in C(K) and x, y E K. We define the generalized translation operator Tx by

Tx(f)(y) = i f(z)d(ox * Oy)(z).

Properties. We have the following properties:

(i) Tx(f)(y) = Ty(f)(x), for all f E C(K) and x, y E K.

(ii) If fis in Cc(K), then for all xE K, the mapping y ---> Tx(f)(y) belongs to Cc(K).

(iii) V x, Y E K, Tx (1)(y) = l. (iv) V xE K, V f E LP(K, m), p E [1, +00], IITx(f) IIp ~ Ilfllp. (v) For f in LP(K, m), p E [1, +00[, the mapping x ---> Tx(f) is continuous from K into LP(K, m), p E [1, +00[.

Definition 1.2. (i) Let Jl and v be in M(K). The convolution of Jl and v is the

measure fJ * v in M (K) defined by

fJ * v(f) = i Tx(f)(y)dfJ(x)dv(y), fE Cc(K).


(ii) If J-l = gm and 1/ = hm with g, hin L 1 (K, m), then

J-l * 1/ = (gm) * (hm) = (g * h)m

where 9 * h is the convolution of 9 and h given by

9 * h(x) = i Tx(g)(y)h(y-)dm(y).

Proposition 1.1. 1f 9 is in LP(K, m) and h is in Lq(K, m) with p, q E [1, +00], then the function g*h belongs to U(K, m), r E [1, +00], with .! + .! - 1 = .! and we have P q r

(see [8j).

1.3. Fourier transform on hypergroups.

A complex valued function X on K will be called a multiplicative function if X is continuous, not identically zero, and has the property that

v x, Y E K, Tx(X)(Y) = X(x)X(y);

we say that X is hermitian if

v xE K, X(x-) = X(x).

The dual of a commutative hypergroup (K, *) is the space K of all bounded multiplicative and hermitian functions X on K, endowed with the compact open topology. Note that K is nonvoid, since it contains the constant function 1, and we have

sup Ix(x)1 = x(e) = 1. xEK

Definition 1.3. (i) For J-l in M(K), the Fourier transform p, of J-l is defined on K by

188 K. Trimeche

(ii) For f in L 1 (K,m), we have

(fm)l\(x) = [f(x)x(x)dm(x);

we denote the first member }(X) and we call ! the Fourier transform of f.

Properties.

(i) For ft in M(K) the function p, is continuous on Rand we have

where Ilftll = fKdlftl is the norm of the measure ft.

(ii) For ft and l/ in M(K) we have

(iii) For f in L 1 (K, m) the function ! is continuous on R, zero at infinity and we have

\j X E R, 1!(x)l::; Ilf111.

(iv) For fand g in L 1 (K, m) we have

\j XE R, (f * g)I\(X) = !(X)§(X)

(see [8]).

Theorem 1.1. There exists a unique positive Radon measure 7r on R, called the Plancherel measure of K, such that for every fand g in (LI n L 2 )(K, m)

[f(x)g(x)dm(x) = bJ(X)§(X)d7r(X).

Notations: We denote by

• LP(K,7r), p E [1, +00], the space of measurable functions on K such that

/lfllp,1r = (fR /f(X)/Pd7r(X)) l/P, 1 ~ P < +00

and IIflloo,1r = esssup /f(x)/ < +00. xER

• S(K) (resp. S(K)) the space of simple functions on K (resp. K) which vanish outside a set of finite measure.

Theorem 1.2. The mapping f ---t ! from (L1 n L2 )(K, m) into L2 (K,7r) extends to an isometrie isomorphism from L2 (K, m) onto L2 (K,7r) (see [8], p. 41).

Corollary 1.1. (i) Let f be in L1(K, m). Then for alt xE K we have

(ii) Let f be in L1(K, m) and g in L2 (K, m). Then

Theorem 1.3. Let f be in C(K) n Ll(K, m) sueh that f belongs to L1(K,7r). Then we have the inversion formula

f(x) = fR !(X)X(x)d7r(X) , xE K

(see [8], p. 73).

Proposition 1.2. For all fand g in L2(K, m) and for all iJ! in S(K) and () in S (K) we have the relations

(1.1) L f * g(x)iJ!(x)dm(x) = fR!(X)f}(x)lft(X)d7r(X)

(1.2) L f * g(x)(}(x)dm(x) = fR!(X)!J(X)B(X)d7r(X).

190 K. Trimeche

Proof. Let 9 E L2(K, m) and W E S(K). We consider the operators f l and f 2 defined on L2 (K, m) by

fl(f) = LI * g(x)w(x)dm(x)

f 2(f) = k l(x)g(x)~(x)d7r(X). From Proposition 1.1, Corollary 1.1 and Theorem 1.1 we deduce that for all I E (LI n L2 )(K, m) we have

But (LI n L2)(K, m) is dense in L2(K, m). Then to prove that f l == f2 it suffices to show that f l and f2 are continuous on L2(K, m).

Using Proposition 1.1, Hölder's inequality and Theorem 1.1 we obtain

and

Ifl(f)I:::; 11I * glloollwlll :::; 1III1211g11211WII1

If2 (f)I:::; 111· gII1,1T11~lloo,1T :::; 111112,1T1IgI12,1T1I~lIoo,1T

= 1111I2I1gI1211~lloo,1T'

This completes the proof of the relation (1.1). We obtain the relation (1.2) by the same proof as for the relation

(1.1). •

Theorem 1.4. Let land 9 be in L2 (K, m). Then the junction 1* 9 belongs to L2(K, m) il and only il the lunction I· 9 belongs to L2(K, 7r) and we have

(1.3)

Proof. (i) We suppose that 1* 9 E L2(K, m). As land 9 are in L2 (K,m), from Theorem 1.2 the function I· 9 belongs to Ll (K,7r).

Thus for all e E S (K) the function 1 . 9 . e belongs also to V (K, n) From the relation (1.2), Hölder's inequality and Theorem 1.2 we deduce

thus the quantity

is finite. From the converse of Hölder's inequality (see [5], p. 181) we deduce that 1· 9 E L 2 (K,7r). We obtain the relation (1.3) from the relation (1.2), Theorem 1.2 and the density of S(K) in L 2 (K, 7r).

(ii) We suppose that 1· gE L2(K, 7r). As fand 9 are in L2(K, m) then, from Proposition 1.1, the function f * 9 belongs to L'XO(K, m); thus for all W E S(K) the function f * g. W belongs to L1(K, m). From the relation (1.1), Hölder's inequality and Theorem 1.2 we deduce

IL f * g(x)w(x)dm(x)1 ~ 1/1· glb,1r11~1/2,1r = 1/1· gI/2,1rI/ WI/2;

we deduce the results by the same proof as for (i).

Corollary 1.2. For fand 9 in L 2 (K, m) we have

both sides are finite or infinite.

•

Proof. For f * 9 in L 2 (K, m) the relation (1.4) can be deduced from Theorem 1.2. For the other case the two of si des (1.4) are infinite. •

In this work we suppose that the commutative hypergroup K satisfies the following conditions

(1.5) (i) The dual K can be identified with a subset r of Cn , n ~ 1, which contains jRn. We denote by <p)" ). E r, the elements üf K; in particular we denote "o the element correspünding to the character 1. We suppose that für all x E K, the mapping ). ---+ .. (x) is cüntinuüus on r.

192 K. Trimeche

(1.6) (ii) The Plancherel measure d7f is absolutely continuous with respect to the Lebesgue measure of ffin, with density 7f; we denote its support by S. We suppose that Sand 7f satisfy

• For all ).. in S the function 'PA (x) is real.

• 7f is continuous on Sand there exist 'Y E lR such that

\:j ).. E s, 17f()..) I s const. (1 + 11)..11)'"1

where

n

11)..11 2 = L l)..jI2, if ).. = ()..1,"" )..n) ES. j=l

• S contain the point 0, and for all a > 0, we have: aS = S.

• \:j ).. E S\{O}, 7f()..) > 0.

• The function k defined on ]0, +oo[ by

7f(~) k(a) = sup

AES\{O} 7f()..)

is continuous on ]0, +00[. • Notations. Using conditions (1.5) and (1.6) we denote by

• F(J-L)()..) and F(f)()..) respectively the Fourier transform p, and j of J-L and f·

• LP(S, 7f), p E [1, +00], the space LP(R, 7f), P E [1, +00].

2. Examples of hypergroups.

In this section we shall give three examples of hypergroups satisfying the conditions (1.5) and (1.6) of the previous section.

2.1. The double coset hypergroup K = K\G/K

Let G be a noncompact connected real semisimple Lie group with finite center, and let 9 be the Lie algebra of G. Let 9 = k + p be the Cartan decomposition of g. Let A c p be a maximal abelian subspace, A* its (real) dual, AC the complexification of A*. The dimension n of Ais called the real rank of G. We can identify A with lRn. We denote


by 11 . 11 the norm in A and by L; theset of restricted roots. Let W be the Weyl group associated with L; and IWI its cardinality.

Fix a Weyl chamber A+ in A and let A+ be its closure. We call a root positive if it is positive on A+. Let L;+ be the set of positive roots,

and put L;t = L;+ n {o: E L;/~ 0: E L;}. Put P = ~ L: m n .0:, with nE2:+

m n the multiplicity of 0:. Let K (resp. A) be the analytic subgroup of G with Lie algebra k (resp. A), and H : G -t A the Iwasawa projection according to the Iwasawa decomposition G = KAN, i.e. if gE G, then H(g) is the unique element in A such that 9 E K(expH(g))N. On the compact group K the Haar measure dk is normalized by fKdk = 1. In

the Cartan decomposition G = KexpA+K, and the Haar measure dx of G is given by

where f is a continuous function on G with compact support and 8(H) = TInE2:+ [2sho:(H)]m a ; here and throughout sh and ch represent the hyperbolic sine and hyberbolic cosine. We have the following estimate for the density 8(H):

The spherical functions on Gare the functions

4?'\(x) = l e(i,\-p)(H(xk))dk, xE G, A E AC·

They satisfy the following properties

(i) The function 4?'\(x) is a C=-function in x and a holomorphic function in A.

(ii) We have

• V A E Ac, 'P,\(e) = 1

• V xE C, 4?ip(X) = 1.

(iii)We have the product formula

194 K. Trimeche

(iv) We have

Itp,\ (x)1 ::; tpo(x), (x E G, A E A*)

with

for some positive constant a. (See [6], [7].)

Definition 2.1. (i) For all x, y E G and f a continuous function on G bi-invariant under K, we put

The mapping * is called a generalized convolution product on the double coset space K = K\G/K.

(ii) The involution on K is defined by

Theorem 2.1. (i) (K, *) is a hypergroup in the sense of Jewett and is calted a double coset hypergroup.

(ii) The Haar measure on K is given by the relation (2.1). (iii) The dual space can be identified with A* + iCP, where CP is

the convex hult of p defined by

CP = {w· p/w E W}.

(iv) The Plancherel measure is given by

where c is the Harish-Chandra function. (v) We have

S = SUPP7r = A*.

(vi) We have

IC(Aja)I-2 {a-2CardEt, if a 2:: 1 k(a) = sup _ = .

AEA*\{O} IC(A)I 2 a-d1mN , if 0< a < 1.

In particular, when G has a complex structure, we have

k(a) = a-2cardEt, for alt a > O.

(For (i), ... ,(v) see [lj, [6j, [7j, and for (vi) see [16j.)

2.2. The Chebli-Trimeche hypergroups.

We consider the function A defined on [0, +oo[ satisfying the fol-lowing conditions

(i) A(x) = x 2a+1 B(x), where a > -~ and B is a positive even Coo-function on ]R such that B(O) = 1.

(ii) Ais increasing and unbounded.

(iii) ~ is decreasing on [0, +oo[ and limx->+oo ~(~] = 2p 2:: O.

(iv) There exists a positive constant 1'0 such that for all r E [a, +00[, a> 0, we have

• If p > 0 . B'(x) - 2p - 2Q.±.l + e-'YO X F(x) . B(x) - x

B'(x) • If p = 0: B(x) = e-'YO X F(x)

where F is a Coo-function on [a, +oo[ bounded together with its derivatives. Let LA be the differential operator on ]0, +00[:

Examples.

(i) If

then LA is the radial part of the Laplacian operator on ]Rk.

196 K. Trimeche

(ii) If

A(x) = (shx)k, k E N, k 2: 2,

then LA is the radial part of the Laplace-Beltrami operator on the k-dimensional hyperbolic space.

We consider the equation

(2.2) {LAU = _(,X2 + p2)u,

u(O) = 1, u'(O) = O.

Proposition 2.1. (i) The equation (2.2) admits a unique COO-solution 'P, which can be extended to an even function on lR.

(ii) For each x E [0, +00[, the function ,X --+ 'P.x (x) is an entire function on C.

(iii) For each x E [0, +00[, we have

1'P.x(x)1 :S 1

if and only if'x belongs to the closure ofthe set ~ = {A E C/llm,X1 < p}. (iv) We have

(See [13], [14], [18].)

Examples.

(i) If

we have

v x E [0, +00[, 'Pip(X) = 1.

1 et> --

2

if 'xx i=- 0

if 'xx = 0

where Ja is the Bessel function of first kind and order et.

(ii) If

1 et > ß >--- 2

we have

where 'P~a,ß) is the Jacobi function defined by

with 2Fl the Gauss hypergeometrie function. (See [13], [14].)

Proposition 2.2. The function 'P>., A E <C, satisfies the following product formula

l X +Y

V x, Y E]O, +00[, 'P>.(x)'P>'(Y) = 'P>.(z)W(x, y, z)A(z)dz Ix-Yl

where W(x, y, z) is a nonnegative continuous function on ]Ix -yl, x+y[ supported in [Ix - yl, x + y], such that

1= W(x, y, z)A(z)dz = 1.

(See [14], [18], [19], [20].)

Examples.

(i) In the case of the normalized Bessel function ja we have

W(x,y,z)

{ aa[(x + y)2 - z2]a-l/2[z2 - (x _ y)2]a-l/2 .

( ) , If Ix - yl < z < x + Y = xyz a

0, otherwise

where

(See [13], p. 93.)

21- 2af(a + 1) aa = J7r f(a + 1/2)'

(ii) In the case of the Jacobi function 'P~a,ß) the function W(x, y, z) is given by

198

• For a > ß > - ~, we have

W(x,y,z) =

1 171"[ 2 2 2 ba ß (h h h)2 1 - eh x - eh y - eh z , s xs ys Z 01 0

K. Trimeche

+ 2ehxehyehz eos Ol~-ß-l

+ (shO) 2ßdO , if Ix - yl < z < x + y

0,

where

and [al+ denotes

{ a,

[al+ = 0,

• For a = ß, a > - ~, we have W(x,y,z) =

r(a + 1) 2J1f r(a + 1/2)

if a:2 0

otherwise.

otherwise

[1 - eh2x - eh2y - eh2 z + 4eh2xeh2yeh2 zla-l/2 x +

(sh2xsh2ysh2z)2a '

if Ix - yl < z < x + y

0, otherwise

(See [13], p. 94.)

Definition 2.2. (i) For all x, y E [0, +00[, and j an even eontinuous function on lR, we put

lX+Y

Ox * Oy(f) = j(z)W(x, y, z)A(z)dz. Ix-yl

The mapping * is ealled a generalized eonvolution product on K = [0,+00[.

(ii) We consider the identity mapping id as the involution on K = [0, +00[.

Theorem 2.2. (i) (K, *) is a hypergroup in the sense oj Jewett called a Chebli-Trimeche hypergroup and denoted ([0, +00[, *(A)).

(ii) The Haar measure on ([0, +00[, *(A)) is given by A(x)dx. (iii) The dual space R can be parametrized by the disjoint union

[0, +oo[ U i[O, p]. (See [1], [2], [17], [18], [20].)

Examples. Particular cases of CMbli-Trimeche hypergroups ([0, +00[, *(A))

are Bessel-Kingman and Jacobi hypergroups which correspond respectively to the functions A(x) = x 2o+ 1 , a > -~ and A(x) =

22(o+ß+l) (ShX)2o+1 (chx)2ß+l , a 2 ß > -~. (See [1], [18].)

Remark. The CMbli-Trimeche hypergroups ([0, +00[, *(A)) are, for different functions A, equivalent to all regular hypergroups ([0, +00[, *) (see [10]).

Proposition 2.3. There exists a junction c : <C --+ <C satisjying

where ,A is the solution oj the equation

such that

(i) The junction ,X --+ Ic('x)1-2 is even on lR, nonnegative, continuous and

c( -'x) = c(,x).

(ii) There exist positive constants N, k1 , k2 such that

(iii) There exist a, b E lR such that

200 K. Trimeche

• For p > 0 and a > - ~, we have

• For p = 0 and a > 0, we have

For (i) and (ii) see [2], [18]; lor the prool 01 (iii) we have used Proposition 3.17 01 [2], which is also Theorem 3.7.201 [18}.

Examples.

(i) In the case of the normalized Bessel function jOl. we have

(ii) In the case of the Jacobi function 'P~OI.,ß) we have

201.+ß+l- iAf(a + l)r(i,X) c( ,X) = -r(-:-:;~--:-( a-+-ß-+-1 -+-i ,x--:-)-::-') f--:(-;-~ --:( a....:....---'-::-ß-'-+-l-+-z-· ,x-:-:-)) .

(See [14].)

Proposition 2.4. (i) The Plancherel measure is given by

(ii) We have

S = SUPP7r = [0, +00[.

(iii) The lunction k(a) defined by

is continuous on [0, +oo[ and satisfies the inequalities

• if p = 0 and a > 0: ~l.l ~ k(a) ~ ~'il' for all a > 0 a a

{ MI < k(a) < M 2 if a :2 1

. I a'Y - - a'Yo ' • if P > 0 and 0: > -2: MI M 2

a'Yo ::; k(a) ::; --;;:t' if 0< a < 1 where MI and M 2 are positive constants, "I = max(1, 20: + 1) and "10 = min(1, 20: + 1). (For i) and ii) see [1], [18].)

2.3. The hypergroup associated with the spherical me an operator

The spherical me an operator R is defined for a continuous function f on ]Rn, even with respect to the first variable, by

R(f)(r,y) = lsn-l f(r17,y + r~)dO"n-I(17,~); (r,y) E]R x ]Rn-I

where sn-I is the unit sphere {(17,~) E]R x ]Rn-I/Tf2 + 11~112 = 1} in ]Rn and O"n-I is the surface measure on sn-I of total mass equal to l.

For (f.L, A) E re x ren-I, let 'P/-L,).. be the function defined by

V (r,x) E]R x ]Rn-I, 'P/-L,)..(r,x) = R(e-i ()",.) cosf.L)(r, x)

where (.,.) is the scalar product on ren-I. Then we have

with

{

n-2 (n) Jn-2 (rz) . 2-2-r - 2 n 2' if rz -=1= 0

]n"22 (rz) = 2 (rz)-2-

1, if rz = 0

and J n"22 is the Bessel function of first kind and order n 22.

The function 'P/-L,).. satisfies the following properties.

Proposition 2.5. (i) For every (f.L, A) E C X cn-I, the function (r, x) -t 'P/-L,)..(r, x) is infinitely differentiable on ]R x ]Rn-I and even with respect to the first variable.

(ii) For every (r,x) E]R x ]Rn-I, the function (p"A) -t 'P/-L,)..(r,x) is entire on re x ren-I.

(iii) We have

1'P/-L,)..(r,x)l::; 1, for all (r,x) E [O,+oo[x]Rn-1

202 K. Trimeche

if and only if (J..L, A) belongs to the set

r = [0, +oo[xRn- 1 U {(i/L, A)/J..L E R, A E Rn- 1 and I/LI :::; IIAII}.

(iv) For all (r,x) ER x Rn-l we have

cpo,o(r, x) = 1.

For every (/L, A) E Cx cn-l, the function CPP,,>. satisfies the product formula'V (r,x), (s,y) in [0, +oo[xRn- 1

cpp,,>.(r, x)cpP,,>.(s, y)

(See [li), [15}.)

Definition 2.3. (i) For all (r,x),(s,y) E [O,+oo[xRn-l, and f a continuous function on Rn, even with respect to the first variable, we put

The mapping * is called a generalized convolution product on K = [0, +oo[ xRn- 1 .

(ii) The involution on K is defined by

(r, x)- = (r, -x), for all (r, x) E K.

Theorem 2.3. (i) (K, *) is a hypergroup in the sense of Jewett, called the hypergroup associated with the spherical mean operator R.

(ii) The Haar measure on K is given by rn-1drdx. (iii) The dual space K can be parametrized by the set r.

Proposition 2.6. (i) The Plancherel measure d7r(J..L, A) is given by

with

where 1[o,+00[xIRn-1 and 1[O,II>'lIlxlRn-1 are the characteristic functions of the sets [0, +00[xjRn-1 and [0, IIAII] x jRn-1.

(ii) We have

S = SUPP1r

= [0,+00[xjRn-1 U {(iJ.t,A)/J.t E [0,+00[, A E jRn-1

and ° ~ J.t ~ IIAII} .

(iii) The function k(a) defined on ]0, +oo[ by

satisfies

1r(1:!. ~) k(a) = sup a' a

(I',>')ES\{O} 1r(J.t, A)

1 k(a) = --1' for all a > 0. an -

(For (i) and (ii), see fll}, f15}.}

3. Wavelets on hypergroups.

3.1. Wavelets on the hypergroup K

Definition 3.1. We say that a function 9 in L2 (K, m) is a wavelet on K if there exists a constant G 9 such that

(i) ° < Gg < +00 (ii) For almost all A in S, we have

Gg = IF(g)(aAW-· 100 da

° a

Theorem 3.1. Given a > ° and 9 a wavelet on K in L2 (K, m). Then

204 K. Trimeche

(i) The function A ---t F(g)(aA) belongs to L2(S,1f) and we have

IIF(g)(a.)1I2,1r ::; (k~:)) 1/211g112

where

1f(A/a) k(a) = SUp (A).

).ES\{O} 1f

(ii) There exists a function ga in L2(K, m) such that

'V A E S, F(ga)(A) = F(g)(aA)

and we have

Proof. (i) From the conditions (1.5) and (1.6) we obtain by change of variables

or

We deduce the result from these relations and Theorem 1.2. (ii) Theorem 1.2 gives the result. •

Proposition 3.1. (i) There exists a unique function (tt, t > 0, in L2(K, m) such that

(ii) The function


is a wavelet on K and we have

Proof. We deduce these results from condition (1.6), Theorem 1.2 and Definition 3.1. •

Theorem 3.2. Let 9 be a wavelet on K in L 2 (K, m). Then jor all a > ° and b E K, the junction

(3.1)

is a wavelet on K in L 2 (K, m) and we have

where n, b E K, are the generalized translation operators given in Definition 1.1.

Proof. From Theorem 3.1 the function ga belongs to L2(K, m). Then from the properties of the operator n the function ga,b belongs to L2(K,m).

Using Corollary 1.1 (i) we obtain

F(ga,b)(>') = '1').,(b)F(g)(a>.), a.e. on S;

thus

From Definition 3.1 we have

The properties of the function '1')., and Theorem 1.2 imply that the function ga,b satisfies the conditions of Definition 3.1. •

Proposition 3.2. Let 9 be a wavelet on K in L2(K, m) such that the junction a -t ga is continuous jrom ]0, +oo[ into L 2 (K, m). Then the junction (a, b) -t ga,b is continuous jrom ]0, +oo[xK into L2 (K, m).

206 K. Trimeche

Proof. Let (aa, ba), (a, b) in ]0, +oo[xK. From property (iv) of the translation operator Tb we deduce

We now obtain the result from the continuity of the function a -> ga from ]0, +oo[ into L2 (K, m) and the fact that the function b -> n(gao) is continuous from K into L2 (K, m). •

3.2. Continuous wavelet transform on K.

Definition 3.2. Let 9 be a wavelet on K in L2 (K,m). We define the continuous wavelet transform on K for 1 in L2 (K, m) by

Cf>g(f)(a, b) = [1(X)9a'b(X)dm(x), for all (a, b) E]O, +oo[xK.

This relation can also be written in the form

Cf>g(f)(a, b) = 1 * 9a(b)

where * is the convolution product given by Definition 1.2.

Proposition 3.3. Let 9 be a wavelet on K in L2 (K, m). (i) For 1 in L2 (K, m) we have

V (a, b) E]O, +oo[xK, lCf>g(f)(a, b)1 ~ (k~~)) 1/211/1121IgI12'

(ii) For all 1 in Lq(K, m), q E [1,2], the mapping b -> Cf>g(f)(a, b) belongs to U(K, m) with r E [1, +00] satislying ~ = i - ~, and we have

Proof. (i) From Definition 3.2 and Proposition 1.1 we have

and from Theorem 3.1 we have

thus

(ii) We deduce the result from Definition 3.2 and Proposition 1.1.

• The following theorems are Plancherel and Parseval formulas for

the continuous wavelet transform on K.

Theorem 3.3. Let g be a wavelet on K in L2 (K, m). (i) Plancherel formula for g'

For all f in L2 (K, m) we have

IIfll~ = Cl { (OO lg(f)(a, bW da dm(b). 9 JK Jo a

(ii) Parseval formula for <l>g. For all f1, h in L 2 (K, m) we have

1 1 1100 da fl(X)h(x) dm(x) = -C <l>g(fr)(a, b)g(h)(a, b) - dm(b). K 9 K 0 a

Proof. (i) From Definition 3.2 and the Fubini-Tonelli theorem we have

We deduce from Corollary 1.2

208 K. Trimeche

then from Fubini-Tonelli's theorem we have

The result follows from Theorems 1.2, 3.1 and Definition 3.1. (ii) We deduce the result from (i).

Corollary 3.1. Let 9 be a wavelet on K in L2(K, m) such that for all a > 0 the function 9a is positive. Then for all f in L2 (K, m), we have the inversion formula for the transform <1>g:

1 1100 da f(·) = -C <1>g(f)(a, b)9a,b(·)- dm(b) 9 K 0 a

weakly in L2(K, m).

Proof. From Theorem 3.3 (ii) and Definition 3.2 we have for all h in L 2 (K,m)

[f(X)h(X) dm(x)

= ~g [100 <1>g(f)(a, b) ([ h(x) 9a,b(x)dm(X)) daa dm(b) ,

but from Fubini-Tonelli's theorem and Theorem 3.3 (ii) we have

{ {OO ( I <1>g (f)(a, b)llh(b)19a,b(X) da dm(b)dm(x) lKlo lK a

:::; <1>g(lfl)(a, b)<1>g(lhl)(a, b)- dm(b) < +00. 1100 da

K 0 a

Then from Fubini's theorem

i f(x)h(x)dm(x)

= i (~g i 100 <1>g(f)(a, b)9a,b(X) ~a dm(b)) h(x)dm(x)

and the result follows. •

By Theorem 3.3 the continuous wavelet transform 9 on K is an isometry of the Hilbert space L 2 (K, m) into the Hilbert space L 2 (]0, +oo[xK, Cl da dm(b)) (the space of square integrable func-

9 a

tions on ]0, +oo[xK with respect to the measure Cl da dm(b)). For 9 a

the characterization of the image of 9 we interpret the vectors 9a,b, (a,b) E]O,+oo[xK, as a set of coherent states in the Hilbert space L 2(K,m).

Definition 3.3. A set of coherent states in a Hilbert space 1t is a subset {ge}eE'c of 1t such that

(i) L is a locally compact topological space and the mapping C --t

ge : L --t 1t is continuous. (ii) There is a positive Borel measure dC on L such that, for f in

1t,

Theorem 3.4. Let {ge}eE'c be a set of coherent states in a Hilbert space 1t. Define the isometry of1t into L 2 (L, dC) (the space of square integrable functions on L with respect to the measure dC) by

(f)(C) = (f, ge), f E 1t.

Let F be in L 2 (L, dC). Then F belongs to (1t) if and only if

(See [9], p. 37-38.) Now let 1t = L 2 (K, m), L =]0, +oo[xK. Choose a wavelet 9 on K

in L 2 (K, m) such that the function a --+ 9a is continuous from ]0, +00] into L 2 (K, m) and let ge = 9a,b be given by Theorem 3.2 if C = (a, b) E L. Then we have a set of coherent states. Indeed, (i) of Definition 3.3 is satisfied because of Proposition 3.2, and (ii) of Definition 3.3 is satisfied for the measure Jg :a dm(b) on ]0, +oo[xK. (See Theorem

3.3 (i).)

Theorem 3.5. Let 9 be the continuous wavelet transform on K, with 9 a wavelet on K in L 2 (K, m), such that the function a --t 9a is

210 K. Trimeche

continuous fram ]0, +oo[ into L2 (K, m). Let F be in L2 (]0, +oo[xK, Cl da dm(b)). Then there exists a function f in L2 (K, m) such that

9 a

if and only if

F(a, b) = ~g [100 F(a', b') ([ gal,bl(X)ga,b(X) dm(x)) ~~' dm(b').

Proof. Apply Theorem 3.4 with 1i = L2 (K, m), C =]0, +oo[xK, coherent states ga,b and measure dC given by Jg :a dm(b). •

Theorem 3.6. Let 9 be a wavelet on Kin L2 (K, m). For f a function in C(K) n L 1(K, m) (resp. C(K) n L2 (K, m)) such that F(f) belongs to L 1 (S,1!') (resp. (LI n LOO)(S,1!')) we have the following inversion formula for g:

(3.1) f(x) = ~g 100 ([ g(f)(a, b)ga,b(x)dm(b)) daa

where, for each x E K, both the inner integral and the outer integral are absolutely convergent, but possibly not the double integral.

Proof. We put

i(a, x) = [g(f)(a, b)ga,b(x)dm(b)

and

(3.2) 1 100 da I(x) = -C i(a,x)-; g 0 a

we shall prove that for each x E K, the integrals i(a, x) and I(x) are absolutely convergent and we have

I(x) = fs F(f) (>-.)cp>. (x)1!'(>-')d>-'.

(i) We suppose that f E C(K) n L 1 (K, m) such that F(f) E L1(S,1!'). From Theorem 3.2 and Definition 3.2 we have

Proposition 1.1 and the properties of the operator Tx imply that the functions b ~ f * 9a(b) and b ~ Tx(ga)(b) belong to L 2 (K, m); then from Hölder's inequality the integral i( a, x) is absolutely convergent. On the other hand, from Corollary 1.1 we have

F(f * 9a)(>") = F(f) (>")F(ga) (>..), a.e. on S

and

then using Theorem 1.2 we obtain

(3.3) i(a, x) = 1s F(f)(>..)IF(ga)(>")1 2.(x)7f(>")d>".

Thus

but from Definition 3.1 we have

(3.4) 1 1= da -C IF(gaW- = 1. 9 0 a

Then

(3.5) 1 1= da -C li(a, x)l- :S IIF(f)lk7r < +00. 9 0 a

From this inequality we deduce that the integral I (x) is absolutely convergent.

We now prove the relation (3.2). From the relation (3.3) we have

From the inequality (3.5) we deduce that we can apply Fubini's theorem. Then we have

212 K. Trimeche

we obtain the relation (3.2) from the relation (3.4). We deduce relation (3.1) from Theorem 1.2.

(ii) We suppose that f E C(K) n L2 (K, m) such that F(f) E (L l n LOC)(S, n) From Theorem 1.4, the function f * [ja belongs to L 2 (K, m) and we have

The same proof as for (i) gives the relation (3.2). We deduce relation (3.1) from Theorem 1.2. •

Remark. Wavelets and continuous wavelet transforms on the double coset hypergroup K = K\G/K, on Chebli-Trimeche hypergroups and on the hypergroup associated with the spherieal mean operator are studied in [4], [15], [16], [17].

References

[1] W.R. ßLOOM-H. HEYER: Harmonie analysis of probability measures on hypergroups, de Gruyter Studies in Mathematies, Vol. 20, Editors: H. Bauer-J.L. Kazdan-E. Zehnder, de Gruyter, Berlin-New York, 1994.

[2] W.R. ßLOOM-Z. XV: The Hardy-Littlewood maximal function for Chebli-Trimeche hypergroups, Contemporary Math., Vol. 183, 1995, pp. 45-69.

[3] C.K. CHVI: An introduction to wavelets, Academic Press, 1992. [4] A. FITOVHI-K. TRIMECHE: J. L. Lions transrnutation oper

ators and generalized continuous wavelet transform, Preprint. Faculty of Sciences of Tunis, 1995.

[5] G.ß. FOLLAND: Real analysis: modern techniques and their applications, John Wiley & Sons, New York, Chiehester-BrisbaneToronto--Singapore, 1984.

[6] R. GANGOLLI-V.S. VARADARAJAN: Harmonie analysis of spherical functions and real reductive groups, Vol. 101, Springer-Verlag, Berlin-New York, 1988.

[7] S. HELGASON: Group and geometry analysis" integral geometry, invariant differential operators and spherieal functions, Academic Press, New York, 1984.

[8] R.I. JEWETT: Spaces with an abstract convolution of measures, Advances in Maths. 18, 1975, pp. 1-101.

[9] T .H. KOORNWINDER: Wavelets: An elementary treatment


of theory and applications, Edited by Tom H. Koornwinder, Series in Approximations and Decompositions, Vol. 1, World Scientific, 1993.

[lOJ Y. MEYER: Ondelettes et operateurs I. Actualites Mathematiques, Hermann, Editeurs des Sciences et des Arts, 1990.

[l1J M.N. NESSIBI-L.T. RACHDI-K. TRIMECHE: Ranges and inversion formulas for spherical me ans operator and its dual, J. Math. Anal. and Appl., Vol. 196, 1995, pp. 861-884.

[12J A.L. SCHWARTZ: Classification of one-dimensional hypergroups, Proceedings of the A.M.S., Vol. 103, No. 4, 1988, pp. 1073-1O8l.

[13J K. TRIMECHE: Transformation integrale de Weyl et theoreme de Paley-Wiener associes a un operateur differentiel singulier sur (0, +00), J. Math. Pures et Appl., Vol. 60, 1981, pp. 51-98.

[14J K. TRIMECHE: Transrnutation operators and mean periodic functions associated with differential operators, Math. Reports, Vol. 4, No. 7, 1988, pp. 1-282; Harwood Academic Publishers, London-Paris-N ew York-Melbourne.

[15J K. TRIMECHE: Inversion of the spherical mean operator and its dual using spherical wavelets, Proceedings of the conference held in Oberwolfach, Oct. 23-29, 1994, Editor: H. Heyer, World Scientific, Singapore-New Jersey-London-Hong Kong, 1995.

[16J K. TRIMECHE: Continuous wavelet transforms on semisimple Lie groups and on Cartan motion groups, C. R. Acad. Sei. Canada, Vol. XVI, No. 4, 1994, pp. 161-165.

[17J K. TRIMECHE: Inversion of the J. L. Lions transrnutation operators using generalized wavelets, Preprint. FacuIty of Sciences of Tunis, 1995.

[18J Z. XU: Harmonie analysis on Chebli-Trimeche hypergroups, Ph.D. thesis, Murdoch University, AustraIia, 1994.

[19J H.M. ZEUNER: One dimensional hypergroups, Adv. Math., Vol. 76, No. 1, 1989, pp. 1-18.

[20] H.M. ZEUNER: Limit theorems for one-dimensional hypergroups, Habilitations-Schrift, Tübingen, 1990.

Faculty of Sciences of Tunis, Department of Mathematics, Campus 1060, Tunis, TUNISIA

Semigroups of Positive Definite Functions and Related Topics

M artin E. Walter Dedicated to K.R. Parthasarathy

Abstract

Every locally compact group, C, has defined on it a semigroup of continuous, positive definite functions, P(C). This semigroup, additionally equipped with a normalized, partially ordered, convex structure is a complete invariant of the underlying group. This semigroup has an identity and we investigate what it means to differentiate in the classical calculus sense at this identity. This leads us to the concept of a semiderivation. We are also naturally led to consider the cohomology of continuous, unitary representations of G, as well as the "screw functions" of J. von Neumann and 1. J. Schoenberg, a Levy-Khinchin formula, and a characterization of groups with property (T).

Introduction

This is a slightly modified version of a talk prepared for The International Conferenee on Harmonie Analysis held in Delhi, India, in December of 1995.

If G is a locally compact group, a complex-valued function, p, defined on G is said to be positive definite if for each choice of natural number n = 1,2,3, ... and for each choice of n elements, g1,.··, gn,

1991 Mathematics Subject Classijication. Primary 43A30, 22A30j Secondary 46L99.

Key words and phrases. C*-algebra, convolution, completely bounded, duality, Fourier-Stieltjes algebra, locally compact group, positive definite function, matrix entry, unitary representation.

216 M.E. Walter

from the group, the n by n complex matrix

is positive definite (note that the diagonal elements of this matrix are all p(e), where e is the identity of the group G).

Let P( G) be the set of continuous, positive definite functions on G. Let P(Gh = {p E P(G) : p(e) ~ I}. Replacing P(G) by P(Gh is what I call normalization. P( Gh is a convex set; if PI, P2 E P( Gh, then )..PI + (1 - )..)P2 E P(Gh for 0 :S ).. :S 1. P(Gh is also partially ordered in the sense that if p, q E P( G) I, then P ~ q if and only if q - P is positive definite, Le., q - P E P( G). In this paper we will use the symbol ~ for the partial ordering relation on P(G). Thus, for example, if 0 is the zero function, 0 ~ P for all P E P( G).

If p,q E P(Gh, then the pointwise product, pq, is also in P(GhThis follows from the fact that pq(g) = p(g)q(g) for gE G, and the fact that the Schur-Hadamard product of two positive definite matrices is again positive definite, see [7], Lemma D.12, p. 683. Note that if (aij) and (bij ) are two matrices then the Schur-Hadamard product of these two matrices is (aij) 0 (bij ) = (aij bij ).

Thus, P(Gh is a partially ordered, convex semigroup. If P(Hh is another such partially ordered, convex semigroup for a group H, a map'ljJ : P(Gh -+ P(Hh will be called an isomorphism in this context, if as a map 'ljJ is one-to-one and onto, 'ljJ is affine, I multiplicative, i.e., 'ljJ(pq) = 'ljJ(p)'ljJ(q), for p, q E P(Gh, and order-preserving, i.e, S The following result answers affirmatively a problem posed in [14] about 20 years ago.

Theorem 1. 'Ij; : P(Gh -+ P(Hh is an isomorphism if and only if there exists a topological isomorphism () : H -+ G.

Proof. It is not hard to prove this theorem if you use [2]. For an interesting example with detailed structure worked out see [14]. Let us now give a reasonably detailed proof of Theorem 1. Given () : H -+ G, it is quite straight forward to construct the isomorphism 'Ij; : P( Gh -+

P(Hh. We leave this for the reader. Assurne 'Ij; is an isomorphism of P(Gh onto P(Hh. Without rela

beling 'ljJ it can be linearly extended to all of P( G), by 11 p 11 'Ij;( i&rr) =

IThis means 'Ij;()..Pl + (1 - )..)P2) = )..'Ij;(pI) + (1 - )..)'Ij;(P2), für 0 :S ).. :S 1 and

Pl,P2 E P(Gh·

Semigroups of Positive Definite Functions 217

'I/J(P), for all nonzero pE P(G). The finite linear combinations of elements from P( G) form a com

mutative Banach algebra, B(G), called the Fourier-Stieltjes algebra of G, cf., [5], [13). The Fourier-Stieltjes algebra of G is related to other algebras "on" G. Recall that there is associated with every 10-cally compact group G the Banach *-algebra L 1(G) of functions (modulo the usual almost everywhere equivalence relation), absolutely integrable with respect to left Haar measure. Addition in LI (G) is pointwise, convolution is the product and the involution is given by f#(g) = fj.-l(g)l(g), where l(g) = f(g-I). The overbar, as usual, denotes complex conjugation.

This Banach *-algebra LI (G) has what is referred to as a universal, enveloping C*-algebra, denoted C*(G), see [4], §13.9, for details. If things are getting a bit too abstract for the reader, take a break from this proof and see what all this means in the more concrete case of finite groups, discussed immediately after this proof.

The Fourier-Stieltjes algebra, B (G), is isometrically linearly isomorphie with the linear (Banach) space of bounded linear functionals on C*(G). The double dual, [13), of C*(G), which is also the dual of B(G), is a von Neumann algebra, W*(G). Thus B(G) is the predual of W*(G).

With the above preliminaries out of the way, let us go back and linearly extend 'I/J from P(G) to all of B(G). First observe that every function b E B(G) can be written in a unique way as a linear combination

of two "self-adjoint" elements, viz., b = (b-t/) +i (b;;'). Thus if we can linearly extend 'I/J to the self-adjoint elements of B(G), we can extend 'I/J to all of B( G). Now any self-adjoint element in B( G) is of the form p-q, where p, q E P(G). Define'I/J on this element as 'I/J(p-q) = 'I/J(P) -'l/J(q). We must verify that this extension is well-defined. But this folIows, sinceifp-q =p'-q', forp,p',q,q' E P(G), thenp+q' =p'+q E P(G). Thus 'I/J(P) + 'I/J(q') = 'I/J(p + q') = 'I/J(p' + q) = 'I/J(p') + 'I/J(q). Hence 'I/J is well-defined on self-adjoint elements, since 'I/J(p) - 'I/J(q) = 'I/J(p') - 'I/J(q').

The above arguments can be applied to the inverse of 'I/J. Thus, 'I/J is a linear isomorphism of B(G) onto B(H). It is easy to verify that 'I/J is also a multiplicative map. Now, it is well-known that a positive linear mapping between preduals of von Neumann algebras is (norm) continuous, but a proof of this fact can be found in [2], Lemma 3.l. Applying this result to 'I/J and the inverse of 'I/J we have that both of these maps are norm continuous.

Taking the transpose of 'I/J, t'I/J : W*(H) --t W*(G), we have a bicontinuous, positivity-preserving map that maps the spectrum of the commutative Banach algebra B(H) C W*(H) into the spectrum ofthe

218 M.E. Walter

eommutative Banaeh algebra B(G) c W*(G). Thus t'IjJ(e) is a positive element (with norm at most one) of the speetrum of B(G), [13]. Thus by a result of Russo and Dye, cf. [11], Corollary 2.9, both t'IjJ and its inverse are norm deereasing maps. Thus 'IjJ is an isometry. It thus follows by arguments in [13], pp. 29-32, that the groups G and H are topologieally isomorphie. •

Let us take a look at finite groups, which are trivially loeally eompaet. There are some highly non-trivial and interesting aspeets to the finite group ease. For a finite group G, L 1(G) and C*(G) eoincide as algebras but have distinet norms.

Suppose G = {gI, ... , gn} is a group with n elements, and suppose that gl = e, the identity. Write the group multiplieation table symmetricaIly, and for simplieity suppose that the extreme upper left entry in this table is g1 1 gl. Thus

where the diagonal entries are all e.

-1 1 gl : gn

-1 9n gn

Any function 1 on G ean be represented as the n by n eomplex matrix obtained by applying the function 1 to eaeh entry in the multiplication table for G; thus 1 is represented by the matrix [1(g;l gj )]. In this representation, eonvolution multiplieation of functions beeomes matrix multiplieation and the involution beeomes the usual eonjugate transpose. Thus LI (G) and C* (G) ean both be represented as the *algebra of all eomplex n by n matrices arising from this eonstruetion. In the first ease the norm is 11 1 1/1 = Ei,j 11(g;1 gj) 1 and in the seeond ease it is the operator norm of the matrix aeting on Cn .

One ean find an n by n unitary matrix, U, ealled the FourierPlaneherel unitary, sueh that UC*(G)U* = tfJf=IMni' where Mni is the ni by ni eomplex matrices, i.e, C* (G) is the direet sum of matrix algebras.

If G is a general loeally eompaet abelian group, then C* (G) is isomorphie with Co(f), the eontinuous, eomplex-valued functions that vanish at infinity on f, the dual group of eontinuous eharacters of G. In this ease P( Gh identifies with the positive (regular, BoreI) measures on f with total mass 1 or less, as weIl as the positive, linear functionals on C* (G) with norm one or less. This last identifieation holds in general, namely, P( Gh identifies with the positive linear funetionals on C* (G)

Semigroups 01 Positive Definite Functions 219

of norm one or less, via the equation

(p,w(J)) = J p(g)/(g)dg,

where f E LI(G) and w(J) is in C*(G), w being the universal representation. Note that P(Gh also identifies with the positive linear functionals on LI (G) that arise via integration with respect to measurable functions. See [4], 13.4.5.

We have the following fundamental definitions (see [15]).

Definition 1. The (generalized) translation of P(Gh by Po E P(Gh is the map p E P( Gh ~ PoP E P( Gh, which is defined since P( Gh is a semigroup.

Definition 2. The translate of a E C*(G) by pE P(Gh, denoted ap, or Tpa, is the unique element in C* (G) that satisfies (ap, q) = (a, pq) for all q E P(Gh.

In [15] we show that Tp is a completely positive map of C* (G) into itself and hence satisfies the Kadison-Cauchy-Schwarz inequality:

Note that ~ is the usual operator order in a C* algebra. This inequality says that Tp defines a non-negative, bilinear form on C*(G) for each p, and it would be interesting (if we had time) to pursue some consequences of that fact.

We now introduce a simple-minded not ion of differentiation at the identity in P(Gh: If {Pn}nEN C P(Gh is an arbitrary sequence converging weakly to 1, we consider the limit limn --+oo Pnl~~1 for each gE G. Thus n

Definition 3. A semitangent vector at 1 E P( Gh is any continuous, complex-valued function 'IjJ on G satisfying 'IjJ(g) = limn --+oo n(Pn(g)-l), for each gE G and some {Pn} C P(Gh, {n} the natural numbers. Note that given a semitangent 'IjJ, defined by {Pn} as above, it is implicit in the definition that 'IjJ(g) exists as a complex number for each 9 E G, and hence that limn --+oo Pn (g) = 1 for all 9 E G. If a semitangent at 1 is 0 at e, we say it is a normalized semitangent. We denote the set of all normalized semitangents by No(G).

We are immediately led to the following characterization of semit angents on G.

220 M.E. Walter

Theorem 2. Let 7jJ be a continuous, complex-valued function on G. Then 7jJ is a (normalized) semitangent at 1 E P(Gh, i.e., 7jJ E No(G), if and only if7jJ(e) = 0, 7jJ(g-l) = 7jJ(g) for alt gE G, andfor each choice of natural number n and each choice of n elements gl, ... ,gn from G the n by n matrix {7jJ(gi 1gj) -7jJ(gi1 ) -7jJ(gj)} is positive hermitian, z.e.,

n

(1) L {7jJ(gi 1gj) -7jJ(gi 1 ) -7jJ(gj)}Ai Aj 2: 0 i,j=l

for any choice of complex numbers Al,"" An.

Proof. The following calculation takes place in the weak closure of C* (G) in its universal representation. This closure is called the universal enveloping W* -algebra, W* (G), and it contains a copy of G.

Suppose limn -+ oo n(Pn(g) - 1) = 7jJ(g) for all 9 E G, where Pn E

P( Gh for all n and Pn (e) = 1 for all n. (If necessary, let p' n (g)~: i!~ . Then limn -+ oo n(p' n -1) = 7jJ -7jJ( e) is a normalized semitangent.) Now consider the set of completely positive maps {Tpn}~=l of W*(G) into itself. Let x = 2::=1 Akgk. Then

= lim n{Tpn (x*x) - x*x + x*x - (Tpnx*)x n-+oo

+ (Tpnx*)x - (Tpnx*)(Tpnx)

= lim n(Tpn - Td(x*x) + lim n(T1 - Tpn)x*x n-+CX) n---+oo

= lim Tn(Pn- 1) (x*x) - lim (Tn(Pn-1)x)*x n-+oo n-+oo

- lim (Tpn xTn(Pn- 1)x). n-+oo

Now for our choice of x = 2::=1 Akgk we see that

m

m

= L )..kAi7jJ(g/:lgi)g/:lgi. i,k=l


In a similar fashion,

For x = 2::;=1 Akgk the above limits can be viewed as being taken with respect to the norm in W*(G), e.g.,

as n ----; 00, for 9 E G. N ow the norm limit of positive elements in W* (G) is positive; hence

we have

m

L ).kAd'ljJ(g-;;1 gi ) - 'ljJ(g-;;1) - 'ljJ(gi)}g-;;1 gi 2 o. i,k=1

Since 1 E P(Gh we have that

thus

m

L ).k Ai{'ljJ(g-;;1 gk ) - 'ljJ(g-;;1) - 'ljJ(gd} 2 0 i,k=1

for each m, each choice of g1, ... ,gm in G and each choice of complex numbers Al, ... ,Am.

We now turn to proving the converse. If the conditions on 'ljJ of Theorem 2 really mean that 'ljJ is a semitangent at 1 in P(Gh, then in analogy with the theory of Lie groups it should not be unexpected that {et,ph2:o is a (pointwise continuous) one-parameter semigroup in P( C) 1. If we could indeed establish this, then 'ljJ must be a semitangent, since 'ljJ = limt_o (1 jt) (et,p -1), pointwise, and et,p E P( Ch for all t 2 o.

We will thus show that a continuous function 'ljJ satisfying equation (1) of Theorem 2 and 'ljJ(g-1) = 'ljJ(g) for all 9 E C, determines

222 M.E. Walter

a one-parameter semigroup of continuous functions of positive type, {et,ph~o. Since 'lj;(e) = 0, this semigroup is in P(Gh. The proof is straightforward. We merely must verify that for t > 0 that et,p is indeed of positive type, i.e.,

n

L '\'\i exp(t'lj;(gj1 gi )) ~ 0 i,j=l

for each natural number n, each choice of complex numbers Al, ... , An and each choice of gl, ... ,gn in G.

Thus

n

i,j=l n

= L exp(t{'lj;(gj1 gi ) - 'lj;(gj1) i,j=l

- 'lj;(gd}) [Ai exp( t'lj;(gi))] [,\j exp( t'lj;(gj))]

2 0,

since if A = ('lj;(gj1 gd - 'lj;(gjl) - 'lj;(gi)) is a positive, hermitian, n x n matrix, the "Schur exponential" etA == 1+ tA + t 2 A 0 A + t 3 A 0 A 0 A + ... is also positive hermitian for t 2 O. Note that we used the fact that 'lj;(g-l) = 'lj;(g) for all 9 E G. •

Corollary. Let'lj; be a continuous function. Then'lj; is a (normalized) semitangent vector at 1 in P(Gh, i.e., 'lj; E No(G), if and only if {et,ph~o is a one-parameter semigroup in P(Gh, with {et,p(e) = 1 for alt t.

It is interesting to note that an alternative characterization of 'lj; E No(G) is as a positive form on an ideal in the *-algebra, Cc(G), the continuous, complex-valued functions on G with compact support. Note that Cc(G) is a *-subalgebra of L1(G). Thus'lj; E No(G) if and only if 'lj;(e) = 0, 'lj;P = 'lj;, and Je Je 'lj;(y-1x)f(y)f(x)dydx ~ 0 for each fE Cc(G) such that Je f(x)dx = O.

We are naturally led to the following definition.

Definition 4. A linear operator 8 defined on a norm dense subspace of C*(G), which we denote Dom(8), with values in C*(G) is called a


semiderivation on C*(C) if xE Dom(8) implies x* E Dom(8) and

8(x*x) 2: (8x*)x + x*8x

whenever x E Dom(8) and x*x E Dom(8).

d2 The prototypieal example of a semiderivation is D = dx 2 on the

realline. In this case we have

D(ll) 2: (Df)f + f(DJ),

where 2: stands for pointwise inequality of functions. In physics, D is an example of a completely dissipative operator. We remark that the equation defining semiderivations defines a nonnegative, bilinear form that warrants furt her investigation.

One might wonder if any higher derivative is a semitangent, or semiderivation, on say ]Rn. The answer is no. The Fourier transform of Dis _x2 . This is a special case of the Levy-Khinchin formula on ]Rn,

whieh says that the semitangents on p(]Rn h are given by

- 7jJ(y) = c + iL(y) + Q(y)

+ r [1 - exp( -i{xIY)) - i(xIY) 2] [1 + 11 X2112 ] dJL(x)

JIRn_{o} 1 + 11 x 11 11 x 11

where x, y E ]Rn, C 2: 0, L is a continuous linear form, Q is a continuous, nonnegative quadratic form and JL is a nonnegative bounded measure on ]Rn - {O} such that the above integral converges, cf. [3J.

Erie Larsen arrived at an analogous formula for the semitangents on P(Ch (see [8]). The main idea is to express a semitangent (or negative definite function) as a (weighted) integral over extreme rays of the convex co ne of all negative definite functions, using the theory of Choquet.

Recall the following definition of a seminorm on a group.

Definition 5. A nonnegative, subadditive function p on G is called a seminorm on C, i.e., p : G --t [0,00) and p(gh) ~ p(g) + p(h), where g,h E G.

There are more seminorms than semitangents, as we can see from the following proposition. In fact, every non-compact, compactly generated group has unbounded seminorms, but some such groups (with property (T)) have only bounded semitangents, 7jJ. In the following Gd is the group G with the discrete topology.

224 M.E. Walter

Proposition 1. Ij'ljJ E No(Gd), then P'lj; = 1'ljJ1! is a seminorm on G.

Proof. This result follows almost immediately by taking the determinant of the following matrix (which is positive definite):

[ 'ljJ(e) - 'ljJ(g) - 'ljJ(g) 'ljJ(h-1g) - 'ljJ(h) - 'ljJ(g)] . 'ljJ(g-lh) - 'ljJ(g) - 'ljJ(h) 'ljJ(e) - 'ljJ(h) - 'ljJ(h)

• J. von Neumann and LJ. Schoenberg studied the following problem,

see [9], [12]. What (semi)metrics p on lR, the realline, exist such that (lR, p) as a metric space is imbeddable isometrically in a (real) Hilbert spaee? In particular, they were interested in finding screw junctions F on lR so that p(x, y) = F(x - y) for x, y E lR. It turns out that this problem has an elegant solution for any locally compact group G. First let us make a formal definition.

Definition 6. A function F is a screw junction on locally compaet group G if (G, p) is, as a metric space, isometrically imbeddable into a (real) Hilbert spaee, where p(g, h) = F(h-1g) for g, hE G.

Proposition 2. A nonnegative, real-valued junction F is a screw junction on locally compact group G ij and only ij - F 2 E No (G).

The proof of this result is not difficult, given the techniques now available, cf. [15]. What is interesting to note is that the screw functions of von Neumann and Schoenberg are so closely related to semitangents and, as it turns out, cohomology.

Cohomological considerations naturally lead us to the possible cocycles whieh can implement an isometrie imbedding of G into some real Hilbert space. See [15] for details of how this is done.

Before stating our last theorem we need at least one more definition.

Definition 7. The function, 1, which is identically 1 on G is a continuous, irreducible unitary representation of G of dimension 1 on Hilbert spaee. Let G be the collection of all unitary equivalence classes of continuous, topologically irreducible representations of G on Hilbert spaee with the Fell topology, see [4], §18.1.5. If {I} is an open set in G, then Gis said to have (Kazhdan's) property (T).

One might guess that if G has property (T) that all derivatives at 1 would be trivial, and this is the ease if trivial means bounded. The groups SL(n,lR) for n 2: 3 have property (T), for example, so such groups do exist.

Semigroups 0/ Positive Definite Functions 225

In the following theorem H 1 (G, H(7r)) is the first (continuous) cohomology group of continuous, unitary representation 7r of G. If this does not mean anything to you, see [15]. For now let me elose this presentation with a theorem from [1).

Theorem 3. Let G be a locally compact, (j-compact group. The following are equivalent: (1) G has property(T); (2) Every 1jJ E No(G) is bounded as a function on G; (3) Every semiderivation ä,p, induced on C*(G) by a 1jJ E No(G), is bounded as an operator on C* (G); (4) H 1 (G,H(7r)) = 0 for all continuous unitary representations 7r of G.

Acknowledgements. We would like to thank the referee for paying careful attention to detail and making several helpful suggestions which improved the exposition of this paper.

Final Note. Since writing this paper, we have discovered an explicit process for recovering a (locally compact) group G from P(Gh. This not only provides another proof of our Theorem 1 above, but it finally provides "the" nonabelian duality theory that most elosely resembles the van Kampen-Pontriagin duality theory for (locally compact) abelian groups. This work is currently in preparation.

References

[1] C.A. Akemann and M.E. Walter, Unbounded negative definite /unctions, Canadian Journal of Mathematics 33 (1981), 862-871.

[2] Wolfgang Arendt and Jean DeCanniere, Order isomorphisms 0/ Fourier algebras, J. Funet. Anal. 50 (1983), 19-143.

[3] C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups, Springer-Verlag, New York, 1975.

[4] J. Dixmier, Les C*-algebres et leurs representations, Cahiers Scientifiques, Fase. 29, Gauthier-Villars, Paris, 1964.

[5J P. Eymard, L'algebre de Fourier d'un groupe localement eompact, Bull. Soe. Math. Franee 92 (1964), 181-236.

[6] Edwin Hewitt and Kenneth A. Ross, Abstract Harmonic Analysis, Vol. 1, Springer-Verlag, Berlin, 1963.

[7J Edwin Hewitt and Kenneth A. Ross, Abstract Harmonic Analyis, Vol. 2, Springer-Verlag, Berlin, New York, 1970.

[8J Eric Richard Larsen, Negative Definite Functions on Locally Compact Groups, Ph.D. Thesis, University of Colorado, Boulder, Colorado, 1982.

[9] J. von Neumann and I.J. Sehoenberg, Fourier Integrals and metric geometry, Trans. Amer. Math. Soe. 50 (1941), 497-251.

[lOJ K. R. Parthasarathy, Multipliers on locally compact groups, Leeture Notes in Mathematies, No. 93, Springer-Verlag, Berlin, New York, 1969.

226 M.E. Walter

[l1J Vern 1. Paulsen, Completely bounded maps and dilations, No. 146, Pitman Research Notes in Mathematical Series, New York, 1986.

[12J 1.J. Schoenberg, Metrie spaces and positive definite functions, Trans. Amer. Math. Soc. 44 (1938), 522-536.

[13J Martin E. Walter, W'" -algebras. and nonabelian harmonie analysis, J. Functional Analysis 11 (1972), 17-38.

[14J Martin E. Walter, Duality Theory for Nonabelian Loeally Compact Groups, Symposia Mathematica, XXII, (1977), 47-59.

[15J Martin E. Walter, Differentiation on the Dual of a Group: An Introduction, Rocky Mountain Journal of Mathematics 12 (1982), 497-536.

Email: [email protected]; Department of Mathematics, Campus Box 395, University of Colorado, Boulder, Colorado, U.S.A. 80309

Charaeters, Bi-Modules and

Representations in Lie Group Harmonie

Analysis

N. J. Wildberger

Abstract This paper is a personal look at some issues in the representation theory of Lie groups having to do with the role of commutative hypergroups, bi-modules, and the construction of representations. We begin by considering Frobenius' original approach to the eh aracter theory of a finite group and extending it to the Lie group setting, and then introduce bi-modules as objects intermediate between characters and representations in the theory. A simplified way of understanding the formalism of geometrie quantization, at least for compact Lie groups, is presented, which leads to a canonical bi-module of functions on an integral coadjoint orbit. Some meta-mathematical issues relating to the construction of representations are considered.

1. Introduction

This paper describes an approach to non-commutative harmonie analysis on a Lie group Gwhich is based on an old idea of Frobenius. We discuss the possible role of commutative hypergroups (in the same vein as [19], [24]), introduce G bi-modules and cOllsider a computational approach to the construction of irreducible representations. Most of the ideas are of an elementary nature. Along the way certain amusing but perhaps unsettling philosophie al points are raised.

This is a personal approach- the opinions expressed are those of the author, and occasionally diverge from the mainstream view. While there is then the increased likelihood of saying something foolish, there is the advantage of encouraging discussion. Good natured debate is a sign of vigour in a theory, and it is in this spirit that I offer this paper.

Harmonie analysis on a non-commutative finite group G was initiated one hundred years aga by G. Frobenius in a paper entitled 'Über Gruppencharaktere' [8]. In this fundamental work Frobenius introduced the idea of a character and computed some character tables.

228 N. J. Wildberger

Nowadays we consider characters useful objects associated to representations, which were not mentioned in Frobenius' 1896 paper. They were introduced by hirn shortly thereafter when the modern definition appears: one starts with a representation 7r : G --+ Gl(n) and defines the associated character to be the function X(g) = tr 7r(g). Our attitude is captured by the following quote of G. Mackey [13].

Frobenius' original definition of character was a complicated one, which emerged from his analysis of Dedekind's problem. A year later, however, he showed that his definition is equivalent to another that is much simpler and more natural.

In other words, the fundamental quest ion today is not

Question 1. What are all the characters of a group G?

as it was for Frobenius in 1896 but rat her

Question 3. What are all the equivalence c1asses of irreducible representations of G?

In my opinion, Frobenius' original point of view towards characters deserves to be reconsidered as possibly more fundamental than the subsequent modern approach. From this standpoint, Question 1 becomes the natural starting point for harmonie analysis on a group G and Question 3 logically follows it. Between these is another reasonable quest ion which seems to have attracted little attention.

Question 2. What are all the equivalence c1asses of irreducible G bimodules?

Frobenius' 1896 approach can be extended to other families of groups by utilizing the concept of a locally-compact commutative hypergroup and possible generalizations. The central object of interest is the class hypergroup lC( G) of conjugacy c1asses of a group G.

For a compact Lie group G, lC(G) is a locally compact, compact hypergroup in the sense of Dunkl [7], Jewett [11] and Spector [15]. There is a basic relation between this hypergroup and the hypergroup of adjoint orbits lC(g) in the Lie algebra gof G given by the wrapping map (see [6]). From this vantage point many important aspeets of harmonie analysis are seen in a new and simplified light. In particular, the classifieation of irreducible representations by highest weights or integral eoadjoint orbits, the Kirillov character formula [12], a formula of Harish-Chandra on

Characters, Bi-Modules and Representations 229

G-invariant differential operators, the Dufio isomorphism, the PoissonPlancherel formula of M. Vergne [18] and the identity of Thompson [17], amongst other things, can be explained both naturally and easily.

For non-compact Lie groups, the definition of IC( G) is more subtle and problematic. At this point, little is known about the right way of proceeding in general although there is evidence that a suitable form of the wrapping map should apply in some situations. What seems clear, however, is that the present theory of locally compact hypergroups is too narrow to accomodate objects which arise as quotients of non-compact group actions. We need to understand hypergroups based on spaces which are not necessarily Hausdorff or locally compact. Its seems a project of some importance to establish such a theory and apply it to the harmonie analysis of non-compact Lie groups. Such a development is also likely to be of interest to mathematical physicists.

In Seetion 2, we define hypergroups, describe Frobenius' original approach to characters, and indicate how to extend this to a compact Lie group. In Section 3, we introduce the not ion of G bi-modules and their potential role as intermediate objects between characters and representations. In Section 4 the natural occurence of such G bi modules for a compact Lie group G as spaces of functions on coadjoint orbits and the connection with moment maps and geometrie quantization is described. The last section raises the possibility that the problem of 'constructing' representations of groups has been obscured by the lack of precision in usage of words like 'construct'. I propose a reasonably concrete meaning for this term and suggest that the task of constructing the irreducible representations of a compact Lie group is by no means completed.

Some problems of interest are scattered through the paper.

2. Hypergroups and Frobenius' approach to characters

The concept which clarifies Frobenius' paper is that of a finite commmutative hypergroup. This not ion is almost implicit in Frobenius' 1896 work. In retrospect, it seems curious that in a century of mathematics oriented towards abstract algebra, this important theory has been developed only recently. Since we will be interested in applications to Lie groups we give a more general definition without apreeise discussion of the topologies involved- see [11] or [4] for more detail.

Definition 1. A locally compact commutative hypergroup is a locally compact space IC for which the Borel measures M(IC) form a *-algebra


satisfying essentially the following axioms. (1) (Closure) The product of Dirac delta functions Dx x Dy , for

x, y E K, is always a compactly supported probability measure which varies continously with x and y.

(2) (Associativity) The algebra M(K) is associative. (3) (Existence of an Identity) There exists an element e E K such

that De is the identity. (4) (Existence of Inverses) For every x E K there exists a unique

element x* E K such that e is contained in the support of the measure Dx x Dx*. Furthermore (Dx )* = Dx*.

(5) (Commutativity) The algebra M(K) is commutative.

In the case of a finite set K = {Co, Cl,···, cn }, this definition coincides, once we identify measures and functions, with the definition given in the paper [16] in this same volume. A finite commutative hypergroup is as close to being a commutative group as possible given that we only require the product of two elements to be a probability distribution of elements; in particular a commutative group is also a commutative hypergroup. The notions of character, duality and Fourier transform for commutative groups extend to finite commutatve hypergroups once we consider also somewhat more general objects called signed hypergroups which involve negative probabilities.

Frobenius' original approach may now be restated into modern language as folIows. For any non-commutative finite group G, there is an associated finite commutative hypergroup K(G), called the class hypergroup of G, obtained from convolving G-invariant probability measures supported on conjugacy classes. The characters of the hypergroup K( G) form a signed hypergroup K( G)/\ which happens to be a hypergroup. The elements of this dual hypergroup are functions on K( G) and so can be naturally interpreted as central functions on G. They are precisely the irreducible characters in the usual sense, except that they have been normalized to have value 1 at the identity.

A number of facts about characters of finite groups such as the orthogonality relations and integrality of character values are special cases of more general facts which hold for finite commutative hypergroups (for some deeper examples, see [2], where the terminology 'table algebras' are used). This is useful since there are many examples of finite commutative hypergroups which arise outside of group theory, for example in the theory of distance regular graphs, association schemes, conformal field theory, cyclotomy, inclusions of Von Neumann algebras etc., see [20].


Summarizing, we may say that the problem of determining the characters of a non-commutative group G is essentially a problem of commutative harmonie analysis; analysis on the associated dass hypergroup K(G).

Let us now try to generalize this approach to the case of Ga compact Lie group. Let K = K(G) be the set of conjugacy dasses of G, which we may view as the quotient of G with respect to the conjugation action on itself. Measures on K( G) form an algebra under convolution induced by the convolution algebra of central measures on G. In this correspondence, delta functions on K( G) are associated to invariant prob ability measures on conjugacy dasses.

A character of K( G) is a bounded continuous function X : K( G) ---+ C

such that (1) X(x)X(y) = JIC(G) X(z)d(ox x Oy)(z) for all x, y E K (2) X(x*) = X(x) for all x E K. The set of characters of a hypergroup K is denoted by K/\. A char

acter of K( G) lifts via the quotient map p : G ---+ K( G) to a function on G invariant on conjugacy dasses.

Definition 2. Any function on G obtained this way from a character of K( G) is called an irreducible normalized character of G.

It is a consequence of a well known theorem of Weyl that this not ion is exactly the same as the usual one; that is, an irreducible normalized character is exactly a function of the form X(g) = ~ tr 7r(g) for some irreducible representation 7r : G ---+ Gl (n).

Our strategy towards understanding harmonic analysis on a compact Lie group G is now the following. The first step is to understand K(G) and its hypergroup structure- this is the basic object. The next step is to determine the characters of K( G) and the hypergroup structure of K( G)/\. The next step is to construct as explicitly as possible the irreducible bi-modules of G and the irreducible representations of G.

The first two steps can be accomplished by turning our attention to the Lie algebra 9 and to certain hypergroup structures on it and its dual g*. These hypergroup structures are special cases of a general phenomenon: for any linear action of G on a vector space V, the space of orbits of G on V carries a hypergroup structure obtained by convolving G-invariant prob ability measures in g.

The resulting orbit hypergroup K(V; G) has dual the orbit hypergroup K(V*; G). Here G acts on the dual vector space as follows:

9 . f ( v) = f (g -1 . v) for all 9 E G, f E V*, v E V.

Furthermore the pairing between the Gorbit U of V and the Gorbit 0 of V' is given by:

(1)

where dJ-lo and dJ-lu are the G-invariant probability measures on 0 and U.

Since G acts on 9 and on g* by the adjoint and coadjoint actions respectively, we have hypergroups K(g ; G) and K(g*; G) which are in duality.

To understand the connection between the dass hypergroup K( G) and the adjoint hypergroup K(g ; G) we now introduce the wrapping map from distributions of compact support on 9 to distributions on G. The consideration of this map is motivated by the work of Harish-Chandra, Helgason, Kashiwara- Vergne, and Dufto. Let j denote a suitably chosen square root of the Jacobian of the exponential map exp : 9 ~ G. This function is analytic on g, has value 1 at 0 and is G-invariant (with respect to the adjoint action). For a function ep on G let 'P = epoexp be its lift to g. For a distribution J-l of compact support on g, define the distribution (J-l) on G by

for any ep E COO(G). The following result, which we call the wrapping theorem, (see [6]) is

a generalization of results of Harish-Chandra, Dufto and I. Frenkel.

Theorem 3. Let J-l and v be two G-invariant distributions of compact support on g. Then

(J-l) * (v) = (J-l * v)

where the convolution on the left is group convolution on G and the convolution on the right is Euclidean convolution in g.

The wrapping theorem allows us to relate the structures of the hypergroups K(G) and K(g ; G). This explains why Kirillov's orbit theory of representations works. Coadjoint orbits and representations are intimately related since the former determine characters of the adjoint hypergroup and the latter determine characters of the dass hypergroup. The character formula of Kirillov follows from the pairing of adjoint and coadjoint orbits. For a more extensive discussion of the applications of this point of view to compact Lie groups, see [6] , [24].

How does any of this generalize to non-compact Lie groups? This is not at all so dear, since the notion of aG-invariant probability measure


on a conjugacy dass in general is not defined, and even if it were, how would one convolve two such things? There are some indications that an appropriate theory of means could be used for some groups at least (see [19]). The fact that much of Kirillov theory extends to non-compact groups suggests that such an approach ought to exist. We know what the answer is- the theory of characters of non-compact Lie groups as developed in the nilpotent case by Kirillov and in the semisimple case by Harish-Chandra and others; the quest ion is - how to recover these results from a hypergroup approach? In other words, how can one obtain the results of representation theory on characters for non-compact Lie groups without mentioning representations? The following is thus a key project.

Problem 1. Develop a more general theory of hypergroups into which the dass hypergroups of non-compact Lie groups may fit.

3. C bi-modules

In this section we introduce a family of objects which perhaps can playa role in harmonie analysis somewhere between characters and representations. The not ion of abi-module is familiar enough in other areas of mathematics, for example the theory of Von Neumann algebras where it has been introduced by Connes.

Let G be a Lie group; that is a smooth manifold with a compatible smooth group structure.

Definition 4. A left-action of C on a finite dimensional vector space V is an assignment to each 9 E C and v E V an element 9 . v E V such that

(1) 9 . v varies smoothly with 9 and v (2) 9 . v varies linearly with v (3) g. (h . v) = (gh) . v for all g, hE C, v E V.

We will also say that C left-acts on V, or that V is a C left-module.

Definition 5. A right-action of C on a finite-dimensional vector space V is an assignment to each 9 E C and to each v E V an element v . 9 E V such that

(1) v . 9 varies smoothly with 9 and v (2) v . 9 varies linearly with v (3) (v· g) . h = v . gh for all g, hE C, v E V.


In this ease we say that V is a G right-module. Notiee that if V is say aG left-module then V* can be made into a G right-module by defining

(f . g) (v) = f (g . v) for all f E V*, 9 E G, v E V.

Definition 6. A vector space W is a G bi-module if it is both a G leftmodule and a G right-module and if these actions are eompatible, that is if

9 . (w . h) = (g . w) . h for all g, h E G, w E w.

Definition 7. A G bi-module W is irredueible if there is no proper subspace Wo c W which is stable under both actions.

Definition 8. A G bi-module W is symmetrie if there exists an involution (a linear map whose square is the identity) * : W ---+ W such that

(g. (w· h))* = (h-1 . w*). g-l

If V is a G left-module then V* is a G right-module and the tensor produet V ® V* is a G bi-module in the obvious way. If in addition we are given a map * : V ---+ V* such that

(g . v)* = v* . g-l

then V ® V* becomes asymmetrie G bi-module by defining

(v ® u*)* = u ® v*.

This is the situation when a G left-aetion on V preserves abilinear symmetrie form ( , ) on V.

Problem 2. For a given group G, eonstruet its irredueible G bi-modules (up to the natural not ion of equivalenee), its irredueible symmetrie bimodules ete.

Remark 1. The requirement of smoothness in the definition of a leftaction is not standard. In the literat ure smoothness is usually replaeed by eontinuity. The reasons are perhaps that

(a) for eompaet Lie groups the two not ions happen to eoincide, and

(b) requiring only eontinuity allows an immediate extension of the definition to infinite dimensional veetor spaees.

Chamcters, Bi-Modules and Representations 235

Quite frankly, this seems to me an unfortunate sleight of hand. Functorial and aesthetic considerations urge us to respect the smooth structure of a Lie group. This means taking care to investigate the meaning of smooth structures on infinite dimensional vector spaces before we extend our definitions to that realm. Since representation theory is of such interest and usefulness, it is appropriate that the basic definitions be natural and functorial.

Remark 2. If one takes a Lie group to mean analytic manifold with analytic group operations, then the above definitions should be modified appropriately. Perhaps harmonic analysis on a smooth Lie group is quite a different subject from harmonic analysis on an analytic Lie group. Note that on the latter, distribution theory is sometimes not naturally available- on a non-compact analytic Lie group for example, there are no naturally occurring functions of compact support!

Remark 3. The terminology of left-action and right-action motivates us to constantly acknowledge the presence of any assymetry which we introduce into the situation. This is a another 'philosophical' point which I believe is of some general usefulness. It is most pleasant when our basic concepts are free of arbitrary bias, even the seemingly innocent one of left vs right. In this context, does anyone know how to define the Lie algebra of a Lie group in a completely symmetric way?

Nevertheless, we will continue to follow standard usage and interchange the terms left-action and representation.

Why should we consider G bi-modules? The reason is that they occur naturally. The most obvious representation of any finite group is the regular representation on the space of all functions on the group. This is of course naturally a G bi-module, since we may translate functions on the left or on the right. As a G bi-module, this space decomposes into irreducible G bi-modules, one for each of the irreducible representations of the group. Each such constituent W is a two sided ideal of the convolution algebra of all functions and contains the character of the representation as an minimal idempotent. As a left-module, W deeomposes into dirn V copies of an irredueible left-module V, but the deeomposition is not eanonieal. This is a reeurring theme in harmonie analysis and quantization theory; the canonical 'square' object which has no natural decomposition into 'left' or 'right' objects. Here we are examining the 'square' objects in their own right.

Given a representation of G on a spaee V, there is an obvious G


bi-module structure on W = End V ~ V ® V·, which is in addition an algebra. Equivalently we may obtain W by considering the space of all matrix coefficients of the representation. The other direction, from W to V, is not at all so easy and is rarely canonical- which is why the bimodule W perhaps qualifies as a simpler object than the representation V.

4. Geometrie quantization revisited

As an illustration of the occurrence of G bi-modules in harmonie analysis, let us return to the situation of a compact Lie group G and consider the method of geometrie quantization which associates to an integral coadjoint orbit 0 an irreducible representation 1[' of G, that is, the Borel-Weil theorem. We briefly review this theory (see [10], [14], [25],) and then sketch a simpler way of understanding it by considering moment maps of representations. The possibility of such a simplification was suggested to the author by a conversation with I. Frenkel. The presence of a canonical G bi-module of functions on 0 follows from this approach.

The assumption that 0 is integral me ans that it carries a complex structure (not unique) such that there exists a holomorphic Hermitian line bundle Lover 0 with connection whose curvature is equal to the canonical symplectic form on 0 (all coadjoint orbits are symplectic manifolds). There is a formula involving the covariant derivative of the connection that shows that the Lie algebra 9 acts naturally on the space of all sections of this bundle. Equivalently one may rephrase this in terms of a circle bundle over 0 and spaces of functions on this bundle that trans form correctiy under the 51 action (see [14] for more details). The irreducible module associated to the orbit is obtained by taking the submodule of holomorphic sections. All irreducible representations of G occur in this way.

The procedure is admittedly somewhat magical, but there is a more conceptual way of understanding it in which the objects and constructions are more natural. The key is to emphasize the inverse procedure of dequantization (see [5], [21]).

Dequantization in this context means going backwards from an irreducible unitary representation 1[' : G -t U(V) to the coadjoint orbit 0 and its associated geometric objects. Here U(V) is the group of unitary

Chamcters, Bi-Modules and Representations 237

operators on some complex inner product space V. Let 0 be the unit sphere of V. We consider the map c/J: 0 -t g* defined by

1 c/J(v)(X) = -:- (d7r(X)v, v)

Z

for v E 0 and X E g. The map c/J is the composition of the projection of 0 onto the projective space PV of V and the moment map of the representation ([9], [21], [22], [23]).

The image of c/J is aG-invariant compact subset of g* whose convex hull has extremal set a single coadjoint orbit O. In fact most of the time the image of c/J is convex , see [1] and [23] for the exact statement. In any case the preimage of this extremal orbit 0 is a single Gorbit M in 0 and c/J : M -t 0 is a circle bundle; the SI action is the restrietion to M of the natural multiplicative action of SI on V. The orbit 0 is the same coadjoint orbit that geometrie quantization utilizes in the construction of 7r; M is actually the orbit of the highest weight vector of the representation and the circle bundle over 0 is the same as that constructed by geometrie quantization. The holomorphic and Hermitian structure of the associated line bundle L follow directly from the holomorphic and Hermitian structure of the space V.

There is furthermore a natural way of assigning to any vector v E V a function Iv on M by the rule Iv(w) = (v,w) for w E U. This function does not push down to 0 but nevertheless has exactly the correct transform properties to identify it with a section of the corresponding li ne bundle L. This realizes V as aspace of seetions of L on which G acts.

We now claim that there is aspace of functions on 0 which carries the G bi-module structure of End V. To an operator T E End V associate the function (JT : M -t IC defined by

(JT(V) = (Tv, v).

Since this function is independent of the phase of v, it is the lift of a function on 0 which we denote by aT. Let Adenote the set of all such functions on O. It is then a fact (see [21]) that the assignment T :-t aT is 1 : 1. Recall that an operator T on a complex vector space is determined by its expectation values (Tv, v) as v ranges over the unit sphere. We are stating that this is still true when the sphere is replaced by the Gorbit M.

Thus we have a canonical identification of End V with the space of functions A on O.


Problem 3. Identify the space A geometrieally.

Problem 4. How does the G bi-module structure of A, which it inherits from the fact that End V is a G bi-module, manifest itself in terms of the geometry of the action of G on O?

A related question, which ties in with the theory of * -products (see [3], [5]) is

Problem 5. How does one describe the algebra structure of A, also inherited from End V as above, in terms of the geometry of O?

Until these quest ions have been answered, the 'construction' of the G bi-module Ais admittedly abstract, but it is of some interest perhaps that we need only functions on the coadjoint orbit, not sections of a line bundle, to exhibit the space. For the case of G = 8U(2) and the irreducible representation of dimension n we may be more concrete. The space A turns out to be all spherical harmonies of degree up to and including n - 1 on the sphere of radius n - 1; aspace of dimension n2•

Problem 6. Develop an analogous theory for the discrete series of a non-compact semisimple Lie group.

5. Construction of Representations

The construction of irreducible unitary representations of a Lie group G is an important problem in non-commutative harmonie analysis. For compact and nilpotent Lie groups, this problem is treated as solved in the literature. It is considered unsolved, for example, for noncompact semisimple groups.

But let us stop for a moment and ask- What does it actually mean to 'construct a representation'? This is a meta-mathematieal question, at least to the extent that one rarely finds a proper definition of the term 'construct' in the literature.

To clarify the discussion, consider once again the case of a compact simple Lie group G. Weyl determined the irreducible characters of such a group in the 1930's. He was certainly aware of the following 'method' of constructing a representation from a character.

(1) Determine the space W of functions on G spanned by all left and


right translates of the given character- this is a G bi-module. (2) Decompose W when viewed as aG left-module into irreducibles- we get dirn V copies of a single irreducible V. (3) Take any one of the constituents-this is the required left-module.

Is this a valid construction? WeIl, no, since the credit for the construction of the representations goes to the Borel-Weil theorem, which only appeared so me decades later. But this is a sociological answer, not a mathematical one. MathematicaIly, one would argue that the instructions (1), (2) and (3) are not specific enough. To what extent is the space W constructed by declaring it to be the span of all left and right translates of a character? How exactly does one decompose this bi-module into irreducibles? Can one exhibit in an explicit fashion some non-zero vector in the final representation space? A basis?

These seem quite reasonable objections, but cannot similar queries be raised about the concreteness of geometrie quantization? How does one explicitly construct a line bundle over so me integral coadjoint orbit? And what does this mean? How does one determine the holomorphic sections of such a bundle? Can one exhibit in an explicit fashion so me non-zero vector of the final representation space? A basis?

There is some scope for disagreement and controversy here. Let us therefore try to define more precisely what we might mean by the metamathematical term 'to construct a representation.'

I tentatively propose the following:

Definition 9. A (finite-dimensional) representation of a group G is constructed if a computer program can be exhibited which will (1) input group elements (in whatever form the group has been 'given') (2) output matrices which represent those group elements in some arbitrary but fixed basis of the representation space.

Of course it is reasonable to require only that explicit instructions for creating such a program be given, not the program itself, at least if we can agree that the instructions are indeed explicit enough. (One should also give some thought to how one 'represents' the real numbers which might appear as the matrix entries). Similarly we will say that one has constructed all the representations of a group when one has a (larger) program which in addition to the above, also inputs the representation (by some label such as the highest weight, or the integral coadjoint orbit 0). One has constructed the representations of all simple compact Lie groups if one has a (yet larger) program that in addition to the above, inputs the group G.


I realize that this proposal will not give universal pleasure. Are there any sensible alternatives? I suspect that some physicists view our abstract constructions of representations as not completely the full storywitness the large number of papers in physics journals concerned with describing explicit bases of representation spaces. It would be unfortunate if young mathematicians miss out on a chance to contribute to this area of physics because of an imprecision of terminology.

If we accept the above tentative definition, some interesting problems present themselves. Primary among them is the following.

Problem 7. Construct the irreducible unitary representations of a compact simple Lie group G.

Here are some others.

Problem 8. Given an irreducible representation of G on V, describe the geometry of the orbits of G on V.

Problem 9. How does one do trigonometry on such orbits?

Problem 10. Describe the hypergroup structure of the G orbits on V.

References

[1] D. Arnal and J. Ludwig, La eonvexite de l'applieation moment d'un groupe de Lie, J. Funct. Anal. 105 (1992), 256-300.

[2] Z. Arad and H. Blau, On table algebras and applieations to products of eharaeters, J. of Alg. 138 (1991), 186-194.

[3] F. Bayen, C. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, Deformation Theory and Quantization 1. Deformations of symplectie struetures, Annals of Physics 111 (1978),61-110.

[4] W. R. Bloom and H. Heyer, Harmonie Analysis of Probability Measures on Hypergroups, De Gruyter, Berlin, (1995).

[5] M. Cahen, S. Gutt and J. Rawnsley, Quantization of Kähler manifolds. 1. Geometrie interpretation of Berezin's quantization, J. Geom. Phys. 7 (1990), no.1, 45-62.

[6] A. H. Dooley and N. J. Wildberger, Harmonie analysis and the global explonential map for eompaet Lie groups, (Russian) Funktsional. Anal. i Prilozhen. 27 (1993), no. 1, 25-32 tranlsation in Functional Anal. Appl. 27 (1993), no. 1, 21-27.

Charaeters, Bi-Modules and Representations 241

[7] C. F. Dunkl, The measure algebra of a loeally eompaet hypergroup, Trans. Amer. Math. Soc. 179 (1973), 331-348.

[8] G. Frobenius, Über Gruppencharaktere, Gesammelte Abhandlungen, Vol. III, Springer-Verlag, Berlin, 1-37.

[9] V. Guillemin and S. Sternberg, Symplectic techniques in physics, Cambridge University Press, 1984, Cambridge.

[10] N. E. Hurt, Geometric Quantization in Action, Mathematics and its Applications Vol. 8, Reidel, 1983, Dordrecht.

[11] R. 1. Jewett, Spaees with an abstract eonvolution of measures, Adv. Math. 18 (1975), 1-1Ol.

[12] A. A. Kirillov, Elements of the Theory of Representations, Grundlehren der math. Wissenschaften 220, Springer-Verlag, Berlia, 1976.

[13] G. Mackey, Harmonie analysis as exploitation of Symmetry, BuH. Amer. Math. Soc. 3 number 1 (July, 1980), 543-698.

[14] J. Sniatycki, Geometric Quantization and Quantum Mechanics, 1980, Springer-Verlag, New York.

[15] R. Spector, Mesures invariantes sur les hypergroupes, Trans. Amer. Math. Soc. 239 (1978), 147-165.

[16] V. S. Sunder and N. J. Wildberger, On diserete hypergroups and their aetions on sets, preprint (1996).

[17] R. Thompson, Author vs Referee: A ease history for middle level mathematieians, Amer. Math. Monthly 90 No. 10 (1983), 661-668.

[18] M. Vergne, A Planeherel formula without group representations, in Operator Algebras and Group Representations Vol. II, Neptune, (1980), 217-226.

[19] N. J. Wildberger, Hypergroups and Harmonie Analysis, Centre Math. Anal. (ANU) 29 (1992), 238-253.

[20] N. J. Wildberger, Finite eommutative hypergroups and applieations from group theory to eonformal field theory, Contemp. Math. 183 (1995),413-434.

[21] N. J. Wildberger, On the Fourier transform of a eompaet semisimple Lie group, J. Austral. Math. Soc. (Series A) 56 (1994), 64-116.

[22] N. J. Wildberger, Convexity and representations of nilpotent Lie groups, Invent. Math. 98 (1989), 281-292.


[23] N. J. Wildberger, The moment map of a Lie group representation, Trans. Amer. Math. Soc. 330 (1992), 257-268.

[24] N. J. Wildberger, Hypergroups, Symmetrie spaees, and wrapping maps, in Prob ability Measures on Groups and related structures, Proc. Oberwolfach 1994, World Scientific, Singapore, 1995.

[25] N. M. J. Woodhouse, Geometrie Quantization, Clarendon Press, 1992, Oxford.

School of Mathematics, University of New South Wales, Sydney, 2052, AUSTRALIA

A Limit Theorem on a Family of Infinite J oins of Hypergroups

Hansmartin Zeuner

Abstract

Let (Sn: n E N) be a random walk on the nonnegative integers with the convolution structure of the join of the subhypergroups {O, ... , n}. It is shown that (for suitable norming constants an --t 0) anSn converges in distribution to a nondegenerate limit if and only if the tail of PSl is a regularly, but not slowly, varying function.

1. Introduction

Limit theorems for random walks on a countably infinite discrete hypergroup have been studied by many authors (Eymard-Roynette [6], Gallardo [8], [9] and Bouhaik-Gallardo [3], [4], Mabrouki [13], Voit [14], [15], [16] and the author [17], [18], [19], [20], [21]). In all of these examples the hypergroups can be defined via sequences of orthogonal polynomials (see [12]), mostly in one variable (except in [3], [4] and [21]). In this article we study the limit behavior of random walks on hypergroups on N that are quite different from the above: We consider a sequence of arbitrary two-element hypergroups Jn = {O, n} for n ~ 1 and define recursively K o := {O}, K n+1 := K n V Jn+1 (the hypergroup join, a hypergroup structure on the set K n +1 = {O, ... ,n + I} defined by Jewett [10], 10.5). By construction, K n is a subhypergroup of K n +1

and so the union K := UnEN K n = N carries a natural (commutative) convolution structure (N, *) having all K n as subhypergroups.

Now let (Sn: n E N) be a randorn walk on K, i.e. So is a.s. the neutral element ° and

for all A ~ K, n E N ,

where J-l is a fixed probability measure on K. In this article we will study the quest ion under which conditions on a norming sequence an --t ° the

Key words and phrases. Centrallirnit theorem, Dornain of attraction Randornized surn, Hypergroup join, Distribution of Maxima.

244 H. Zeuner

random variables anSn converge in distribution to a limit # co. The possible limit laws are quite different from the case of polynomial hypergroups where standard normal, Rayleigh and Gaussian distributions occur; this sterns from the fact that the infinite join K is more elosely related to the max-semigroup on N, where Cm * Cn = Cmax(m,n) for alI m # n and where the limit laws are extreme value distributions too.

We will show in the following that the possible limit distributions have densities x 1--+ cpx-p - 1 exp( -cx-P) on lR+ for some c, p > ° and that the domain of attraction of this limit consists of those fJ E MI (N) such that t 1--+ fJ([t, oo[) = P{SI 2: t} is of regular variation at 00 with index - p. This limit theorem is in a elose relationship with the limit theorem of Fisher and Gnedenko for the distribution of maxima (see FeUer [7], example (b), p. 277). It is remarkable that the possible limit laws and domains of attraction do not depend on the choice of the hypergroups Jn .

2. Notation

Let (K, * K) be a compact hypergroup with normalized Haar measure W K and (J, * J) a discrete hypergroup with neutral element e and involution j 1--+ 3 (see [2) or [10) for the definitions). Then the hypergroup join is the convolution structure on the set K V J := KU (J \ {e}) defined by

Ck*Cj:=Cj*Ck:=Cj forkEK,jEJ\{e},

Cj * C] := 1J \{e} . (Cj *J cl) + (Cj *J C]){e}. WK for jE J \ {e},

and the original convolution of K respectively J in aU other cases. This construction was introduced by Jewett ([10], 10.5), who has shown that we in fact obtain a hypergroup and also has given the foUowing general example of this procedure ([10], example 15.1D):

(2.1). Consider a sequence (bn : n E N) of real numbers with bo = 1 and ° < bn ~ 1 for n 2: 1. For every n 2: 1 let Jn := {O,n} the two-element hypergroup with neutral element ° and Cn * Cn =

bnEo + (1 - bn)En . Then we can define hypergroups K n := {O, 1, ... ,n} recursively for every nE N as the join K n+1 := K n V Jn+1 . Since K n is a subhypergroup of Kn +1 for every n E N we obtain in a natural way a convolution on the infinite join K := N = U~=o K n . Using the above

A Limit Theorem on a Family of Infinite Joins 245

definition of the join we see that Em * En = Emax(m,n) for n -=1= m and

The Haar measure W K on K assigns to every element n E K the mass wK{n} = bn n;~;(1 + b~)·

The hypergroup Ha (with a 2: 1/2) described by Dunkl and Ramirez [5] is the special case with bn = l~a for all n 2: 1 of the definition above. If a = l/p where p E N is a prime number then the infinite join is related to the p-adic group (see [5], §3).

(2.2). The dual k of the infinite join (see [2], 2.2) is homeomorphic to NU {oo}; the characters are given by XOO = IN and

n:Sv

n=v+l

n2:v+2

for n E N. Since XooXv = Xv, XflXV = Xmin(fl,v) for J.L -=1= v, and

00 fl ( ) ,,1 + bV +1 II b", XvXv = 1 - bV+1 Xv + ~ 1 + b 1 b", + 1 Xfl'

fl=v+l fl+ ",=v+l

k is a hypergroup with respect to pointwise multiplication. The Fourier transform for this hypergroup is defined for every probability measure P on N as the function on k = NU {oo} with

FP(v):= J XvdP=P({O, ... ,v})-bv+1P({v}).

(2.3). We will now consider a random walk on the infinite join K with jump distribution J.L := PSI· The distribution of Sn is then PSn = J.L*n := J.L * ... * J.L (n factors). Since Km is a subhypergroup of K for every m E N, it follows from the Kawada-Ito theorem for compact hypergroups (Bloom, Heyer [1]) that the distributions ofthe generalized sums Sn converge to the normalized Haar measure WKm if P{SI > m} = 0 and P{SI = m} > O. This measure satisfies WKm {k} = bk n;:k l';b j for k 2: 1 and WKm {O} = n7=1 l';b j • We are therefore

246 H. Zeuner

only interested in the case where f.l is not supported by finitely many points of N.

3. Limiting distributions and domains of attraction

(3.1) Theorem. Let So, Sl, S2, ... be a random walk on K = N such that the function x J-t P{ Sl ~ x} is of regular variation in the sense of Karamata [11), but not of slow variation.

Then there exist a sequence of positive real numbers an > 0 and p > 0 such that the sequence an . Sn converges to a distribution on R+ with Lebesgue density x J-t p. x- p - 1 exp( -x-P).

Proof. Since x J-t P{ Sl ~ x} is a decreasing function, it varies regularly with exponent -p < 0 (see FeIler [7], pp. 275-277). Choose Sn in such a way that n . P{Sl ~ sn} -t 1 as n -t 00. Then it follows from the definition of regular variation (FeIler [7], (VIII.8.5)) that limn --. oo n . P{Sl ~ XSn} = X- P for all x > O. This implies limn --. oo n· P{Sl ~ XS n + j} = x-P for all JEN since for every c > 0 we have XS n + j :=; (x + c)sn, if n is big enough. From this we obtain n· P{Sl = I xSn 1 + j} -t 0 as n -t 00.

It follows from (2.2) that

Hence for JEN and x > 0 we obtain

For every m E N it follows by inspection of the definition of the characters that

I[O,m]

00 j-1

" 1 II bm +i+1 ~ b ·Xm+j. j=O bm + j + 1 i=O 1 + m+i+1

Since the product is at most 2- j , the convergence is uniform. Further-

more this speed of convergence implies that for every x > °

Since pointwise convergence of the distribution functions implies weak convergence, this completes the proof of the theorem if we set an :=

I/sn. •

We conclude the discussion of this limit theorem on the infinite join hypergroup by showing that the distributions described in the theorem and the remark above are the only possible limit distributions on this structure and that the domain of attraction consists exactly of the distributions described in the theorem.

(3.2) Theorem. Suppose that P{SI < a} < 1 for alt a > ° and that there exists a sequence of real numbers Sn > ° such that Sn/sn converges in distribution to a non degenemte probability measure on IR+.

Then there exist c, p > ° such that the limiting distribution has a Lebesgue density x r--> cp . x- p - l • exp( -cx-P) and the function x r-->

P{ SI 2 x} is of regular, but not of slow variation.

Proof. We show first that the sequence (sn : n E N) converges to infinity. For, if this is not the case, there exists a subsequence (nk : k E N) such that snk converges and hence PSnk converges toward a measure Q E M1(N). This implies that FPS1 (v) --t FQ(v) and therefore FQ(v) E {-I, 0,1} for all v E NU {oo}. Since FQ is continuous at the unit character XOO we have FQ(v) = Q( {O, ... ,v}) -bv+1 Q( {v}) = 1 = FQ( 00) for so me v E N. This implies Q( {O, ... ,v}) = 1 and therefore P{SI :::; v + I} = 1, in contradiction to the assumption.

248 H. Zeuner

Let F be the distribution function of the limiting distribution of Sn/ Sn and let x > 0 be a point of continuity of F. Then we have

ifn 2: no(c). This implies limn -+oo P{Sn = rxsn 1+1} = 0 and therefore

lim (FPs1(fxsnl)t = lim FPsn(fxsnl) = lim P{Sn < rxsnl + I} n-+oo n~(X) n~oc>

= lim P{Sn/sn < x} = F(x). n-+oo

This implies limn-+oon· (1- FPs1(rxsnl)) = -lnF(x) for all x > o where F is continuous. Since F is nondegenerate, it follows from Lemma 3 on page 277 in FeUer [7], that x f--+ 1-F PS1 (r xl) is of regular variation and there exist c > 0 and p 2: 0 such that F(x) = exp( -cx-P). It is clear that also limn -+ oo n(1-FPs1 (rXSn 1 +j-1)) = F(x) for every jE N,x > O.

By using the same decomposition of l[O,m] in terms of the Xm+j

as in the proof of Theorem (3.1) we obtain that limn -+ oo n . P{SI 2: xsn } = exp( -cx-P ) and hence x f--+ P{SI 2: xsn } is of regular variation. Since F is a distribution function, p = 0 is not possible, and the tail probability is not of slow variation. •

References

[1] W. BIoom, H. Heyer, Convergence of convolution produets of probability measures on hypergroups, Rend. Math (3), Sero VII 2 (1982), 547-563.

[2] W. BIoom, H. Heyer, Harmonie analysis of probability measures on hypergroups, de Gruyter, Berlin, New York, 1995.

[3] M. Bouhaik, 1. Gallardo, Une lai des grands nombres et un theoreme limite eentral pour les ehaaines de M arkov sur N2 assocü§es aux polynomes discaux, C. R. Acad. Sei. Paris 310, Sero I (1990), 739-744.

[4] M. Bouhaik, L. Gallardo, Un theoreme limite eentral dans un hypergroupe bidimensionnel, Ann. Inst. Poincare 28,1 (1992), 47-6l.

[5] C. F. Dunkl, D. E. Ramirez, A family of eountable eompact P*hypergroups, Trans. AMS 202 (1975), 339-356.

[6] P. Eymard, B. Roynette, Marches aleatoires sur le dual de SU(2), Analyse harmonique sur les groupes de Lie. Proceedings 1973-1975. ed. by P. Eymard Lecture Notes in Mathematics vol. 497, Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 1975, pp. 108-152.


[7] W. FeIler, An Introduction to Probability Theory and Its Applications, Volume II, John Wiley & Sons, Inc., New York, Chichester, Brisbane, Toronto, Singapore, 2nd edition 1971.

[8] L. Gallardo, exemples d'hypergroupes transients, Probability Measures on Groups VIII, ed. by H. Heyer, Lecture Notes in Mathematics vol. 1210, Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 1984, pp. 68-76.

[9] L. GaIlardo, Gomporlement asymptotique des marches aleatoires associees aux polynomes de Gegenbauer et applications, Adv. Appl. Prob. 16 (1984), 293-323.

[lOJR. 1. Jewett, Spaces with an abstract convolution of measures, Adv. Math. 18 (1975), 1-1Ol.

[11]J. Karamata, Sur un mode de croissance reguliere, Mathematica (Cluj) 4 (1930), 38-53.

[12JR. Lasser, Orthogonal polynomials and hypergroups, Rend. Mat. Ser. VII 3 (1983), 185-209.

[13JM. Mabrouki, Principe d 'invariance pour les marches aleatoires associees aux polynomes de Gegenbauer et applications, C.R. Acad. Sc. Paris, Serie I (19) 299 (1984),991-994.

[14JM. Voit, Laws of large numbers for polynomial hypergroups and some applications, J. Theor. Prob. 3, No. 2 (1990), 245-266.

[15JM. Voit, Gentral limit theorems for a class of polynomial hypergroups, Adv. Appl. Prob. 22 (1990), 68-87.

[16JM. Voit, Gentral limit theorems for random walks on No that are associated with orthogonal polynomials, J. Multiv. Anal. 34 (1990), 290-322.

[17JHm. Zeuner, Laws of large numbers for hypergroups on lR+, Math. Ann. 283 (1989), 657-678.

[18JHm. Zeuner, The centrallimit theorem for Ghebli-Trimeche hypergroups, J. Theoretical Probab. 2 (1989), 51-63.

[19]Hm. Zeuner, Moment functions and laws of large numbers on hypergroups, Math. Z. 211 (1992), 369-407.

[20JHm. Zeuner, Invariance principles for random walks on hypergroups on lR+ and N, J. Th. Probab. 7 (1994), 225-245.

[21JHm. Zeuner, Limit theorems for polynomial hypergroups in several variables, Probability measures on groups and related structures XI, ed. by H. Heyer, World Scientific, Singapore, 1995, pp. 426-436.

Email: [email protected]@math.uni-dortmund.de; Institut für Mathematik der Medizinischen Universität zu Lübeck, Wallstraße 40, D-23560 Lübeck, Federal Republic of Germany

harmonic analysis and hypergroups

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