On the convergence of negative-order Cesàro means of Fourier and Fourier-Walsh series
type: Item
instance of: scholarly article
title: On the convergence of negative-order Cesàro means of Fourier and Fourier-Walsh series
zbMATH Open document ID: 1504.42006
DOI: 10.1134/S0001434622090152
published in: Mathematical Notes
publication date: 2022-10-25T00:00:00Z
review text: Recently it was proved by \textit{M. G. Grigoryan} and \textit{L. N. Galoyan} [J. Math. Anal. Appl. 434, No. 1, 554--567 (2016; Zbl 1331.42004)] that there exists an increasing subsequence of positive integers \(\Lambda = \{M_{\nu}\}_{\nu =1}^{\infty}\) with density \(1\) such that for any function \(f(x) \in L^{0}[-\pi, \pi]\) and any number \(\varepsilon > 0\), one can define a function \(g\) continuous on \([-\pi, \pi]\) with \(\mu \{f\not = g\}<\varepsilon\) and such that the subsequence \(\sigma_{M_{\nu}}^{\alpha}(g, T)\), \(\nu = 0,1, \dots,\) of Cesáro means of order \(\alpha \in (-1, 0)\) of the trigonometric Fourier series of \(g\) uniformly converges on \([-\pi, \pi]\). In the present paper, it is stated that in the case when \(\alpha <-\frac{1}{2}\), \(\alpha \not = -1, -2, \dots,\) this theorem can be strengthened, namely the corrected function \(g\) will have, in addition, uniformly convergent trigonometric Fourier series. The author notes also that the theorem remains true for the Walsh system.
Mathematics Subject Classification ID: 42A20
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 42A16
Mathematics Subject Classification ID: 40G05
zbMATH DE Number: 7606793
zbMATH Keywords: trigonometric Fourier series
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: Walsh system
zbMATH Keywords: Cesàro means
zbMATH Keywords: convergence
author: Martin Grigoryan
MaRDI profile type: Publication
cites work: APPLICATION OF CESÀRO SUMMABILITY METHODS OF NEGATIVE ORDER TO TRIGONOMETRIC FOURIER SERIES OF SUMMABLE AND SQUARE SUMMABLE FUNCTIONS
cites work: PROPERTIES OF CESÀRO MEANS OF NEGATIVE ORDER AND OF CERTAIN OTHER $T$-MEANS FOR FOURIER SERIES OF CONTINUOUS FUNCTIONS
cites work: Q5835540
cites work: On the uniform convergence of negative order Cesaro means of Fourier series
cites work: Application of negative order Cesàro summability methods to Fourier–Walsh series of functions from $L^{\infty }[0, 1]$
cites work: Convergence of Fourier series in classical systems
cites work: Q4183180
cites work: Q4730237
cites work: On the universal and strong \((L^1,L^\infty)\)-property related to Fourier-Walsh series
description: scientific article; zbMATH DE number 7606793