Pointwise convergence of \(n\)-dimensional Hermite expansions
type: Item
instance of: scholarly article
title: Pointwise convergence of \(n\)-dimensional Hermite expansions
zbMATH Open document ID: 0892.42015
author: Mark A. Pinsky
author: Carl L. Prather
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-08-09T00:00:00Z
review text: The authors prove the following theorem. Suppose that an integrable function \(f(x)\), \(x\in\mathbb{R}^n\), is piecewise smooth with respect to \(x=0\). Then the spherical partial sums of the \(n\)-dimensional Fourier-Hermite series of \(f\) converge if and only if the smoothness index (introduced in this paper) satisfies the inequality \(j(f; 0)\geq[(n- 3)/2]\). The authors also prove the following auxiliary result, interesting in itself. Let \(a\neq 0\), then the series \(\sum^\infty_{k=1} k^{-\beta}e^{ia\sqrt k}\) converges if and only if \(\beta> 1/2\).
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 902545
zbMATH Keywords: pointwise convergence
zbMATH Keywords: Hermite expansion
zbMATH Keywords: spherical partial sums
zbMATH Keywords: \(n\)-dimensional Fourier-Hermite series
reviewed by: Ferenc Móricz
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0164
OpenAlex ID: W2092005162
DOI: 10.1006/JMAA.1996.0164
description: scientific article; zbMATH DE number 902545