Uniqueness of rectangularly convergent trigonometric series
type: Item
instance of: scholarly article
title: Uniqueness of rectangularly convergent trigonometric series
zbMATH Open document ID: 0780.42015
DOI: 10.2307/2946621
author: J. M. Ash
author: Chris Freiling
author: Dan Rinne
published in: Annals of Mathematics. Second Series
publication date: 1993-06-29T00:00:00Z
review text: For a vector \(X=(x_ 1,\ldots,x_ n)\in\mathbb{R}^ n\) and (real or complex) coefficients \(a_ M\), \(M=(m_ 1,\ldots,m_ n)\in\mathbb{Z}^ n\), we consider the multiple trigonometric series \(T:=\sum_{M\in\mathbb{R}^ n}a_ Me^{iMX}\). The type of convergence is crucial to the results of the paper: if \(Z=(z_ 1,\ldots,z_ n)\in\mathbb{Z}^ n\) and we denote by \(R_ Z\) the rectangle \(R_ Z:=\{M:| m_ j|\leq| z_ j|\), \(j=1,\ldots,n\}\), then \(T\) is said to be rectangularly convergent when \(\exists\lim_{\min_{j=1,\ldots,n}| z_ j|\to\infty}\sum_{M\in R_ Z}a_ Me^{iMX}\). It was proved by Georg Cantor (1870) that if a one-dimensional trigonometric series converges to zero everywhere, then all of its coefficients must be zero. A discussion is given of various generalizations of Cantor's result, of which the only nontrivial direct analogues are for double series, namely that if a double trigometric series is either rectangularly convergent, or circularly convergent, to zero everywhere, then all of its coefficients must be zero; however, even the corresponding theorem for square convergence of a double trigonometric series is still unknown, which indicates the delicacy of the problem. The present paper deals decisively with the case of rectangular convergence by proving that if a multiple trigonometric series is rectangular convergent to zero at every point, then all of its coefficients are zero. The proof is a tour-de-force: the authors select those ideas of the masters (Cantor, Riemann, Schwarz,...) which are capable of generalization, but add their own substantial innovations. The bibliography is informative.
Mathematics Subject Classification ID: 42B99
Mathematics Subject Classification ID: 40B05
zbMATH DE Number: 221041
zbMATH Keywords: multiple trigonometric series
zbMATH Keywords: rectangular convergence
reviewed by: D. C. Russell
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2946621
OpenAlex ID: W2033008848
description: scientific article; zbMATH DE number 221041