Trigonometric Fourier series and Walsh-Fourier series with lacunary sequence of partial sums
type: Item
instance of: scholarly article
title: Trigonometric Fourier series and Walsh-Fourier series with lacunary sequence of partial sums
zbMATH Open document ID: 1272.42017
DOI: 10.1134/S0001434613010367
author: S. K. Bloshanskaya
author: O. V. Lifantseva
author: Igor L. Bloshanskii
published in: Mathematical Notes
publication date: 2013-04-26T00:00:00Z
review text: Let \(\psi\) be a trigonometric or Walsh system on the set \([0,1)\). Let \(k\in\mathbb N\) be a fixed number, \(J_k=\{j_1,j_2,\dots,j_k\}\) be a fixed subset of \(\{1,2,\dots,d\}\), \(\mathbb N_{j_\nu}\) be a lacunary sequence, and \[ \mathbb N^d(J_k)=\mathbb N\times\dots\times\mathbb N\times\mathbb N_{j_1}\times\mathbb N\times\dots\times\mathbb N\times\mathbb N_{j_k}\times\mathbb N\times\dots\times\mathbb N. \] Let \(E\subset[0,1)^d\) and \(\mu E>0\). We say that, for multiple Fourier series with respect to the system \(\psi\) in the class \(L_p([0,1)^d)\), \(p>1\), weak generalized localization almost everywhere takes place on the set \(E\subset[0,1)^d\) with respect to the set \(\mathbb N^d(J_k)\), if for any function \(f\in L_p([0,1)^d)\), \(f(x)=0\) on \(E\), there exists a subset \(E_1\subset E\), \(\mu E_1>0\) such that \[ \lim\limits_{{n\to\infty\atop n\in\mathbb N^d(J_k)}}S_n(f,x,\psi)=0 \] almost everywhere on \(E_1\). The authors find a necessary and sufficient condition on the set \(E\), under which for multiple Fourier series with respect to the system \(\psi\) in the class \(L_p([0,1)^d)\) weak generalized localization takes place on the set \(E\subset[0,1)^d\) with respect to the set \(\mathbb N^d(J_k)\).
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 42B05
zbMATH DE Number: 6158191
zbMATH Keywords: multiple Fourier series
zbMATH Keywords: Walsh-Fourier series
zbMATH Keywords: lacunary sequence of partial sums
zbMATH Keywords: Walsh-Paley system
zbMATH Keywords: Lebesgue measure
reviewed by: S. F. Lukomskii
MaRDI profile type: Publication
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full work available at URL: s0001434613010367
OpenAlex ID: W2092205589
description: scientific article; zbMATH DE number 6158191