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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in $\mathbb{R}^{d}$


Author: Yuan Xu
Journal: Proc. Amer. Math. Soc. 126 (1998), 3027-3036
MSC (1991): Primary 33C50, 42C05, 41A63
MathSciNet review: 1452834
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Abstract: We study the Fourier expansion of a function in orthogonal polynomial series with respect to the weight functions

\begin{displaymath}x_{1}^{\alpha _{1} -1/2} \cdots x_{d}^{\alpha _{d} -1/2}(1-|\mathbf{x}|_{1})^{\alpha _{d+1}-1/2}\end{displaymath}

on the standard simplex $\Sigma ^{d}$ in $\mathbb{R}^{d}$. It is proved that such an expansion is uniformly $(C, \delta )$ summable on the simplex for any continuous function if and only if $\delta > |\alpha |_{1} + (d-1)/2$. Moreover, it is shown that $(C, |\alpha |_{1} + (d+1)/2)$ means define a positive linear polynomial identity, and the index is sharp in the sense that $(C,\delta )$ means are not positive for $0 <\delta <|\alpha |_{1} + (d+1)/2$.


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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/S0002-9939-98-04415-3
PII: S 0002-9939(98)04415-3
Keywords: Orthogonal polynomials in several variables, on simplex, Ces\`{a}ro summability, positive kernel
Received by editor(s): March 14, 1997
Additional Notes: Supported by the National Science Foundation under Grant DMS-9500532.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society