Pointwise convergence of the Fejér means of functions on unbounded Vilenkin groups
type: Item
instance of: scholarly article
title: Pointwise convergence of the Fejér means of functions on unbounded Vilenkin groups
zbMATH Open document ID: 0972.42019
published in: Journal of Approximation Theory
publication date: 2001-11-07T00:00:00Z
review text: Although summability for Vilenkin-Fourier series in the bounded case has been known for nearly a quarter century [see \textit{J. Pal} and \textit{P. Simon}, Acta Math. Acad. Sci. Hung. 29, 155-164 (1977; Zbl 0345.42011)], the unbounded case has defied resolution until now. In this interesting paper, the author proves that if \(f\in L^p(G)\) for \(p> 1\), where \(G\) is \textit{any} Vilenkin group (of bounded type or not), then the Vilenkin-Fourier series of \(f\) is \((C,1)\) summable to \(f\) almost everywhere. The hang-up had been that the Fejér kernels for the unbounded case where \textit{not} uniformly bounded in the \(L^1\) norm. The author gets around this dilemma by using an ingenious decomposition of the maximal Cesàro operator into four pieces which isolate the difficulties one at a time.
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 43A75
zbMATH DE Number: 1389107
zbMATH Keywords: Vilenkin-Fourier series
zbMATH Keywords: Vilenkin group
zbMATH Keywords: Fejér kernels
zbMATH Keywords: maximal Cesàro operator
author: G. Gát
reviewed by: William R. Wade
MaRDI profile type: Publication
full work available at URL: jath.1999.3346
OpenAlex ID: W2085336295
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DOI: 10.1006/JATH.1999.3346
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description: scientific article; zbMATH DE number 1389107