an application of the theory of basic series to theorems of bernstein-widder type

An Application of the Theory of Basic Series to Theorems of Bernstein-Widder TypeAuthor(s): Philip DavisSource: American Journal of Mathematics, Vol. 72, No. 4 (Oct., 1950), pp. 787-791Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2372295 .

Accessed: 29/09/2013 00:42

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access toAmerican Journal of Mathematics.

http://www.jstor.org

This content downloaded from 128.118.88.48 on Sun, 29 Sep 2013 00:42:44 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=jhup

http://www.jstor.org/stable/2372295?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


AN APPLICATION OF TH{E THEORY OF BASIC SERIES TO THEOREMS OF BERNSTEIN-WIDDER TYPE.*

By PHILIP DAVIS.

1. Introduction. That the signs of the derivatives of a function of class C? on an interval may influeiice its allalytic character was first pointed out by S. Bernstein [1]. Bernstein's theorem may be stated as follows.

THEOREM 1. Let f(x) be a real valted function defined and of class C- in 1 < x < 1. If each derivative of f(x) is non-negative in 1 < x < 1, then f(x) is necessarily analytic in that interval.

Some years later, a result of a similar nature, but this time showing the effect of the signs of the even derivatives, was obtained by D. V. Widder [2].

THEOREM 2. If f (x) is real and of class C>- in -1 < x < 1 and if

(1) (-1)f(2n) (x)>O (-1<x<1;n==O,1,2 ),

then f(x) necessarily coincides with an entire function of exponential type at most 7r.

Functions satisfying the condition (1) have been termed completely convex. Subsequently Boas and Polya [3] gave a generalization which con- tained as special cases both Theorems 1 and 2. In the present note these theorems are generalized in another direction by obtaining similar results for a number of familiar sets of differential operators. We shall find it convenient to formulate these results using the notion of the basic series of polynomials introduced by Whittaker [4]. Our work is essentially algebraic in character, inasmuch as all questions are referred back either to Theorem 1 or to Theorem 2.

2. Basic sets and series. In Whittaker's terminology, a set of polynomials {Pn(z) } is called a basic set if an arbitrary polynomial can be expressed in a unique fashion as a finite linear combination of the pn's. Whittaker has given the following necessary and sufficient condition that a set of polynomials be basic.

* Received April 6, 1950. 787



788 PHILIP DAVIS.

THEOREM 3. Given the set of polynomials

(2) Pn(Z) - pnjZj, (n= 0, 1, 2< ) j=o

where in (2) we have inserted dummy zero coefficients; a necessary and sufficient condition that the set (2) be basic is that the row-finite matrix

(3) P (Pni) (n, j 0 , 1~,2,**

have an inverse Q which is row-finite:

(4) PQ QP I.

By a row-matrix is meant one in which each row has but a finite number of non-zero elements. It may also be remarked here that the multiplication of row-finite matrices is associative, so that the above condition is equivalent to the existence of a unique Q for which QP I.1

With a given basic set of polynomials (2) with matrix P = (pnj) and with unique row-finite inverse Q (qnj), there may be associated the following set of (possibly formal) power series:

(5) qn(Z) E qkn Zk/17! (n - O, 1, 2, . . . k=0

The associated set may be characterized in the following way. The set {qn(z) } is associated to the basic set {pn(z) } of polynomials if and only if

00

(6) i pn (z) qn (W) - ew n=0 holds as a formal identity, in the sense that when substitutions are made, order of summation reversed, and coefficients compared, we have equality. it is easily seen that as a formal identity, (6) is equivalent to (4).

We shall give a second characterization of the associated set. For two 00 00

power series p (z) = anZn q (z) = bnzn introduce the inner product n=o n=0

(7) [p, q] E anbn,n!. n=o

If either p or q is a polynomial, there can be no question of the convergence of the series in (7). The set (5) is associated to the basic set (2) if and only if the two sets are biorthonormal:

(8) [pi, qj] = ij.

1 See, e. g., Dienes [5].



AN APPLICATION OF THE THEORY OF BASIC SERIES. 789

With a given function f (of an appropriate class) and a given basic set, there may be associated the following formal expansioii of f:

oo

(9) A Z) E [qnnf]Pn(Z)., n=0

the so-called basic series for f. The study of the representability of functions in a series (9) forms a major chapter in the theory of functions. For our purposes, however, we need only know that equality in (9) holds whenever f is a polynomial. Since {pn (z)} is basic, there is a unique finite expansion of a polynomial fi: f=- cpo + c1p1 + + Ckpk. Hence by (7) and (8), ci [qi, f].

We shall now list a number of basic sets together with their associated sets. Our list is intended to be suggestive rather than exhaustive, and may be augmented at will. Moreover, the examples have been chosen so that the associated functions are expressible in closed form. Except when otherwise noted, the 0-th polynomial is 1.

Maclaurin Polynomials:

(10) Pn (Z) zn/n !; qn(Z) - zn-

Newton Interpolation Polynomials:

pn(Z)= (z ao)(z- a,) ( . (z--a1), as distinct;

1 1 ...1 1 1 ...1

aO a, . . .a. aO a,. a

(11) qn(Z) = . . . aon-1 a-n-1 . . .ann-1

ea&0z e* . . e a z ael a*n

Gontcharoff Polynomials (Whittaker [4]):

(12) Pn(Z) 4 dz1 dz2* . dznp qn(z) =zneanz/n!.

Lidstone Polynomials (Widder [2]):

po (z) = Z, p2n+1 (Z) p2(t ( Z);

(13) p2n-2(Z) p2n(z), p2n(O) P2n(l) ?0

q2n (Z) - q2n+1,z) -



790 PHILIP DAVIS.

Hermite Polynomials: [n/2]

(14) pn (Z) =- (-1) k (2z) n-2k/k!(n-2k) ! qn (Z) ZneZ2/4 /2nn!. k=O

Appel Polynomials (Boas [6]):

A (w) ezw pn (z) wn, qn (z) zn/A (z), n=0

A(w) regular at w=O, A(O) #0.

3. Applications. We mav Ilow state our principal theorem.

THEOREM 4. Let {pn(z) } be a basic set of polynomials wvith real coefficients, and let {qn(z)} be the (possibly formal) set of associated power series. For each n, let [qj(z),Zn] qnj have a constant sign as j varies. Then, with f(x) a real valued function of class C- on - 1 < x < 1,

(16) Pn (D) f (x) 0 (n - 0, 1, 2, * -1 <x < 1; D dldx)

implies that f (x) is analytic in - < x < 1.

Proof. We have, for some integer kn kli kit

Zn I [qj(z), zn]p,(z) ; hence Dnf = E qnjpj(D)f. =0 j=o

The inequality (16) now implies that Dnf (x) has constant sign in -1 < x < 1. The theorem is therefore an immediate consequence of Theorem 1.

It may happen that the associated series qn (z) have non-negative coefficients for all n. In other words, the basic set {pn(Z)} has a positive inverse property. All the sets of our list, with proper restrictions on the arbitrary constants appearing therein, have this property.

COROLLARY 4. 1. If ai ? 0, then the sets (10), (11), (12), (13), (14) have a positive inverse property. This is also true of the set (15) provided that [A(w)]-l has non-negative Mllaclaurin coefficients. For the polynomials of any of these sets, condition (16) implies analyticity in -- 1 < x < 1.

Proof. Inspection of the corresponding q4f's reveals that under the above conditions their coefficients are indeed non-negative. For the set (11) we should add that qnj = aoioali *. ajiJ wlhere the summation is extended over all non-negative i's for which io + i1 + * * * + ij n -j.

Theorem 4 is capable of further generalization, for Theorem 1, as stated, is but a special case of Beriistein's results. Actually, Bernstein has shown



AN APPLICATrION OF rHE rT[EORY OF B3ASIC SERIES. 791

that if each successive difference of f(x) is of constant sign in the interval, then, without assuming the existence of derivatives, it will follow that f(x) is analytic in the interval. If therefore in Theorem 4, pt,(D) be replaced by p,, (A), the conclusion follows without assuming the existence of derivatives.

By selecting the Newton Polynomials (11) and combining Theorems 4 and 2, we arrive at the following generalization of Widder's Theorem.

COROLLARY 4. 2. Let f(x) be real and of class CCO on -1 < x < 1, and suppose that as are real. Thent

(17) f(x) - 0, (- 1)n(D2 + ) . . . (D2 + a "2)f(x) ? 0

(n 1, 2, - 1 < x < 1) implies that f(x) is completely convex on the interval and hence coincides with an entire futnction of exponential type at most 7r.

HARVAIRD UNIVERSITY.

REFERENCES.

[1] S. Bernstein, " Sur a d6finition et les proprit,6s des fonctions analytiques d'iine

variable r6elle," ilIathematische Annalen, vol. 75 (1914), pp. 449-468. [2] 1). V. WViddei, " Completely con1vex fun1Ction1s and Lidstonie series," Transactions

of the American Mathematical Society, vol. 51 (1942), pp. 387-398. [3] R. P. Boas, Jr. and G. P6lya, " Influcenec of the sigins of the derivatives of a func-

tioni onl its analytic character," J)uke Journal of Mathematics, vol. 9 (1942), pp. 406-424.

[4] J. M. Whittaker, Interpolatory Function Theory, Cambridge T'racts No. 33, Cam- bridge, 1935..

[5] P. Dienes, " Notes on linear equations in inflnite matrices," Quarterly Journal of Alathematics, vol. 3 (1932), pp. 253-268.

[6] R. P. Boas, Jr., "Exponential transformns and Appel polynomials," Proceedings of the National Academy of Sciences, vol. 34 (1948), pp. 481-483.



an application of the theory of basic series to theorems of bernstein-widder type

Documents