Norm convergence of double Fourier series on unbounded Vilenkin groups
type: Item
instance of: scholarly article
title: Norm convergence of double Fourier series on unbounded Vilenkin groups
zbMATH Open document ID: 1399.42075
author: G. Gát
author: Ushangi Goginava
published in: Acta Mathematica Hungarica
publication date: 2018-02-13T00:00:00Z
review text: V. I. Shcherbakov [Anal. Math. 10, 133--150 (1984; Zbl 0563.42015)] proved an analogue of the Dini-Lipshitz test for the convergence of Fourier series with respect to multiplicative Vilenkin systems. It looks like \( \omega(1/M_k,f)=o(1/\log M_{k+1}) \) as \( k\to\infty. \) Here the authors extend this result to the convergence of rectangular partial sums of double Fourier series with respect to multiplicative Vilenkin systems. In particular, they prove the convergence (in \( C \) and \( L_1 \)) of those sums if \(\omega_1(1/M_k,f)=o(1/\log M_{k+1}) \) and \( \omega_2(1/M_k,f)=o(1/\log M_{k+1}) \) as \( k\to\infty. \) The result is sharp in the sense that those sums may diverge if both moduli of continuity (in each of the variables) satisfy the conditions from above with \( O \) instead of \( o. \)
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 42B05
zbMATH DE Number: 6837908
zbMATH Keywords: multiplicative Vilenkin systems
zbMATH Keywords: Dini-Lipschitz condition
zbMATH Keywords: convergence
zbMATH Keywords: rectangular partial sums
reviewed by: Alexei Lukashov
MaRDI profile type: Publication
full work available at URL: s10474-017-0703-9
OpenAlex ID: W2598864969
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DOI: 10.1007/S10474-017-0703-9
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description: scientific article; zbMATH DE number 6837908