On the representation of functions by trigonometric series
type: Item
instance of: scholarly article
title: On the representation of functions by trigonometric series
zbMATH Open document ID: 0878.42005
author: T. W. Körner
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 1998-01-08T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73405
review text: The aim of this fascinating paper is to give an account of a number of Menshov's theorems among others, on the adjustment of measurable functions, on the existence of universal trigonometric series, on the limits of indeterminacy of trigonometric series, etc. The present author has developed an approach different from that of Menshov and presents a coherent treatment in the paper under review. Let us write \[ S_n(f,x):= \sum^n_{k=-n}\widehat f(k)e^{ikx} \] and, if \(E\) is a measurable set, let us write \(|E|\) for the Lebesgue measure of \(E\) normalized such that \(|\mathbb{T}|=1\). The key result on which everything else depends is formulated in the following lemma. Given any \(\eta>0\) there exists a \(K(\eta)\) with the following property. Given any \(\varepsilon>0\) one can find a positive function \(f\in A(\mathbb{T})\), that is \(\sum|\widehat f(k)|<\infty\), with the following properties: (i) \(\widehat f(0)=1\), (ii) \(|\widehat f(k)|\leq\varepsilon\) for all \(k\neq 0\), (iii) \(|S_m(f,x)|\leq K(\eta)\) for all \(m\) and \(x\), (iv) \(|\{x:f(x)\neq 0\}|<\eta\). We warmly recommend this paper to everyone who is interested in harmonic analysis.
reviewed by: Ferenc Móricz
Mathematics Subject Classification ID: 42A20
Mathematics Subject Classification ID: 42-02
zbMATH DE Number: 1013858
zbMATH Keywords: Menshov's theorems
zbMATH Keywords: adjustment of measurable functions
zbMATH Keywords: universal trigonometric series
MaRDI profile type: Publication
OpenAlex ID: W2073875552
cites work: Q5338374
cites work: Q3218611
cites work: Q5590907
cites work: Q5556050
cites work: Limits of indeterminacy of trigonometric series
cites work: Q4205910
cites work: Q3839914
cites work: Everywhere divergent Fourier series
DOI: 10.5802/AFST.850
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description: scientific article; zbMATH DE number 1013858