On the Lebesgue test for the convergence of Vilenkin-Fourier series
type: Item
instance of: scholarly article
title: On the Lebesgue test for the convergence of Vilenkin-Fourier series
zbMATH Open document ID: 0780.43004
author: D. H. Dezern
author: D. Waterman
published in: Michigan Mathematical Journal
publication date: 1993-06-29T00:00:00Z
review text: In this note the authors adapt Waterman's localization of the Salem test to obtain an extension of the Onneweer-Waterman Lebesgue test for convergence of Fourier series of functions defined on a bounded Vilenkin group. The main result is Theorem 3.3: Let \(G\) be a bounded Vilenkin group, and suppose that \(f\in L^ 1(G)\). Then the Fourier series of \(f\) converges to \(f^*(x)\) at every point \(x\in G\) at which the following conditions hold: (S1) \(f^*(x)\) exists, and \[ \lim_{k\to\infty}{1\over m(G_{k+1})}\int_{G_{k+1}}\sum^{m_ k- 1}_{\alpha=1}{1\over\alpha}\Bigl|\sum^{p _{k+1}-1}_{j=0}f(x- z^{(k)}_ \alpha - jx_ k -t)\zeta^{ja_ k}_ k\Bigr| dt = 0\tag{S2} \] \[ \text{ uniformly in }a_ k\in\{1,2,\dots,p_{k+1}-1\}. \] The convergence is uniform in \(x\) on any closed set of points where \(f\) is continuous and (S2) holds uniformly.
Mathematics Subject Classification ID: 43A50
Mathematics Subject Classification ID: 43A20
Mathematics Subject Classification ID: 42C10
zbMATH DE Number: 222028
zbMATH Keywords: convergence of Fourier series
zbMATH Keywords: bounded Vilenkin groups
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004597
OpenAlex ID: W2042137370
DOI: 10.1307/MMJ/1029004597
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description: scientific article; zbMATH DE number 222028