Summability of Fourier orthogonal series

for Jacobi weight on a ball in

Author:
Yuan Xu

Journal:
Trans. Amer. Math. Soc. **351** (1999), 2439-2458

MSC (1991):
Primary 33C50, 42C05, 41A63

DOI:
https://doi.org/10.1090/S0002-9947-99-02225-4

Published electronically:
February 24, 1999

MathSciNet review:
1475698

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Fourier orthogonal series with respect to the weight function

on the unit ball in are studied. Compact formulae for the sum of the product of orthonormal polynomials in several variables and for the reproducing kernel are derived and used to study the summability of the Fourier orthogonal series. The main result states that the expansion of a continuous function in the Fourier orthogonal series with respect to is uniformly summable on the ball if and only if .

**1.**M. Abramowitz and I. Stegun,*Handbook of Mathematical Functions*, 9th print, Dover Publ., New York, 1970.**2.**R. Askey,*Orthogonal polynomials and special functions*, SIAM, Philadelphia, 1975. MR**58:1288****3.**H. Berens and Y. Xu,*Fejér means for multivariate Fourier series*, Math. Z.**221**(1996), 449-465. MR**97a:42003****4.**H. Berens and Y. Xu,*summability for multiple Fourier integral*, Approximation Theory VIII (C.K. Chui and L. Schumaker, ed.), vol. 2, World Sci. Publ. Co., 1995, pp. 55-62. MR**98e:42012****5.**L. Bos,*Asymptotics for the Christoffel function for Jacobi like weights on a ball in*, New Zealand J. Math.**23**(1994), 99-109. MR**95j:41047****6.**A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi,*Higher transcendental functions*, vol. 2, McGraw-Hill, New York, 1953. MR**84h:33001b****7.**G. Gasper,*Positivity and the convolution structure for Jacobi series*, Ann. of Math.**93**(1971), 112-118. MR**44:1852****8.**G. Gasper,*Positive sums of the classical orthogonal polynomials*, SIAM J. Math. Anal.**8**(1977), 423-447. MR**55:5925****9.**F. John,*Plane waves and spherical means applied to partial differential equations*, Wiley-Interscience, New York, 1955. MR**17:746d****10.**Y. Kanjin,*A convolution measure algebra on the unit disk*, Tôhoku Math. J.**28**(1976), 105-115.MR**53:1178****11.**T. Koornwinder,*Two-variable analogues of the classical orthogonal polynomials*, Theory and Application of Special Functions (R. A. Askey, ed.), Academic Press, New York, 1975, pp. 435-495. MR**53:5967****12.**T. Koornwinder,*The addition formula for Jacobi polynomials and spherical harmonics*, SIAM J. Appl. Math.**25**(1973), 236-246. MR**49:10938****13.**Szeg\H{o}, G.,*Orthogonal polynomials*, 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Providence, RI, 1975. MR**51:8724****14.**Y. Xu,*On multivariate orthogonal polynomials*, SIAM J. Math. Anal.**24**(1993), 783-794. MR**94i:42031****15.**Y. Xu,*On orthogonal polynomials in several variables*, Special functions, -series, and related topics, Fields Institute Communications Series, vol. 14, 1997, pp. 247-270. MR**99a:33009****16.**Y. Xu,*Christoffel functions and Fourier series for multivariate orthogonal polynomials*, J. Approx. Theory**82**(1995), 205-239. MR**96h:42021****17.**A. Zygmund,*Trigonometric Series*, 2nd ed., Cambridge Univ. Press, Cambridge, 1968. MR**38:4882**

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Additional Information

**Yuan Xu**

Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

Email:
yuan@math.uoregon.edu

DOI:
https://doi.org/10.1090/S0002-9947-99-02225-4

Keywords:
Orthogonal polynomials in several variables,
Jacobi weight on the unit ball,
summability,
positive sums

Received by editor(s):
August 17, 1995

Published electronically:
February 24, 1999

Additional Notes:
Supported by the National Science Foundation under Grant 9302721 and 9500532.

Article copyright:
© Copyright 1999
American Mathematical Society