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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Summability of Fourier orthogonal series
for Jacobi weight on a ball in $\mathbb{R}^{d}$


Author: Yuan Xu
Journal: Trans. Amer. Math. Soc. 351 (1999), 2439-2458
MSC (1991): Primary 33C50, 42C05, 41A63
Published electronically: February 24, 1999
MathSciNet review: 1475698
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Abstract | References | Similar Articles | Additional Information

Abstract: Fourier orthogonal series with respect to the weight function
$(1-|\mathbf x|^{2})^{\mu - 1/2}$ on the unit ball in $\mathbb{R}^{d}$ 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 $(1-|\mathbf x|^{2})^{\mu -1/2}$ is uniformly $(C, \delta )$ summable on the ball if and only if $\delta > \mu + (d-1)/2$.


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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: http://dx.doi.org/10.1090/S0002-9947-99-02225-4
PII: S 0002-9947(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