Pointwise convergence of multiple Fourier series using different convergence notions
type: Item
instance of: scholarly article
title: Pointwise convergence of multiple Fourier series using different convergence notions
zbMATH Open document ID: 0598.42012
DOI: 10.1007/BF01908456
published in: Analysis Mathematica
publication date: 1986-01-01T00:00:00Z
review text: Convergence in Pringsheim's sense of a multiple series follows from its regular convergence, but the converse is not true in general. In this paper we study the relation between the convergence notions applied to multiple Fourier series. The following two theorems are typical, where \(T^ d=\{x\in R^ d:\) \(0\leq x_ j<2\pi\) for \(j=1,2,...,d\}\) denotes the d-dimensional torus. Theorem 1. If \(f\in L^ p(T^ 2)\) for some \(p>1\), then the regular convergence and the convergence in Pringsheim's sense of the double Fourier series of f are equivalent, up to a set of measure zero. The situation is sharply different from \(d=3\) on. Theorem 3. There exists a function \(g\in C(T^ 3)\) whose triple Fourier series converges in Pringsheim's sense on a set E of positive measure, but fails to converge regularly a.e. on E.
Mathematics Subject Classification ID: 42B05
zbMATH DE Number: 3964642
zbMATH Keywords: multiple Fourier series
zbMATH Keywords: convergence in Pringsheim's sense
author: Ferenc Móricz
author: Vilmos Totik
MaRDI profile type: Publication
cites work: On the divergence of multiple Fourier series
cites work: Q5769940
cites work: On the convergence in a restricted sense of multiple series
cites work: Some remarks on the notion of regular convergence of multiple series
cites work: Q3254471
full work available at URL: https://doi.org/10.1007/bf01908456
OpenAlex ID: W139258624
description: scientific article; zbMATH DE number 3964642