Divergence of decreasing rearranged Fourier series
type: Item
instance of: scholarly article
title: Divergence of decreasing rearranged Fourier series
zbMATH Open document ID: 0913.42030
DOI: 10.2307/2118587
published in: Annals of Mathematics. Second Series
publication date: 1999-05-30T00:00:00Z
review text: In the present paper, the author shows that there exists a real function \(f\in L^2(\mathbb{T})\) such that \[ \limsup_{\eta\to 0+} \Biggl| \sum_{|\widehat f(k)|\geq \eta} \widehat f(k)\exp(2\pi ikt)\Biggr|= \infty \] for almost all \(t\in [0,1)\), where \(\widehat f(k)\) is the \(k\)th Fourier coefficient of \(f\). This result strengthens the result on the rearrangement of Fourier series stated by Kolmogorov and proven by Ul'yanov and later much simplified by Olevskij: There exists a real \(f\in L^2(\mathbb{T})\) and a bijection \(\sigma:{\mathbf Z}\to{\mathbf Z}\) such that \[ \limsup_{N\to\infty} \Biggl| \sum_{| k|\leq N}\widehat f(\sigma(k))\exp(2\pi i\sigma(k)t)\Biggr|= \infty \] for almost every \(t\in [0,1)\). To construct the function, the author modifies the proof of Olevskij. The concrete construction is rather involved. Using a probabilistic argument basic building blocks are produced finally to construct a modified Olevskij system.
Mathematics Subject Classification ID: 42C20
zbMATH DE Number: 935305
zbMATH Keywords: rearrangement of Fourier series
author: T. W. Körner
reviewed by: Rupert Lasser
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2118587
OpenAlex ID: W2324202080
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description: scientific article; zbMATH DE number 935305