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June 2019 Uniform convergence of orthogonal polynomial expansions for exponential weights
Kentaro ITOH, Ryozi SAKAI, Noriaki SUZUKI
Hokkaido Math. J. 48(2): 263-280 (June 2019). DOI: 10.14492/hokmj/1562810508

Abstract

We consider an exponential weight $w(x) = \exp(-Q(x))$ on ${\mathbb R} = (-\infty,\infty)$, where $Q$ is an even and nonnegative function on ${\mathbb R}$. We always assume that $w$ belongs to a relevant class $\mathcal{F}(C^2+)$. Let $\{p_n\}$ be orthogonal polynomials for a weight $w$. For a function $f$ on ${\mathbb R}$, $s_n(f)$ denote the $(n-1)$-th partial sum of Fourier series. In this paper, we discuss uniformly convergence of $s_n(f)$ under the conditions that $f$ is continuous and has a bounded variation on any compact interval of ${\mathbb R}$. In the proof of main theorem, Nikolskii-type inequality and boundedness of the de la Vall{\'{e}}e Poussin mean of $f$ play important roles.

Citation

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Kentaro ITOH. Ryozi SAKAI. Noriaki SUZUKI. "Uniform convergence of orthogonal polynomial expansions for exponential weights." Hokkaido Math. J. 48 (2) 263 - 280, June 2019. https://doi.org/10.14492/hokmj/1562810508

Information

Published: June 2019
First available in Project Euclid: 11 July 2019

zbMATH: 07080094
MathSciNet: MR3980942
Digital Object Identifier: 10.14492/hokmj/1562810508

Subjects:
Primary: 41A17
Secondary: 41A10

Rights: Copyright © 2019 Hokkaido University, Department of Mathematics

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Vol.48 • No. 2 • June 2019
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