Remarks on the uniform and \(L^1\)-convergence of trigonometric series
type: Item
instance of: scholarly article
title: Remarks on the uniform and \(L^1\)-convergence of trigonometric series
zbMATH Open document ID: 1240.42015
published in: Acta Mathematica Hungarica
publication date: 2011-02-25T00:00:00Z
review text: \textit{T. W. Chaundry and A. E. Jollife} [Lond. M. S. Proc. (2) 15, 214--216 (1916; JFM 46.0455.03)] proved that if \(a_k\) is a nonnegative sequence tending monotonically to zero, then the series \(\sum_{k=1}^\infty a_k \sin kx\) converges uniformly if and only if \(ka_k\to 0\) as \( k\to\infty\). This was extended by \textit{S. P. Zhou, P. Zhou and D. S. Yu} [Sci. China, Math. 53, No. ~7, 1853--1862 (2010; Zbl 1211.42006)] to more general classes of sequences. The author extends these results to even more general classes of sequences. He also extends a recent result of \textit{S. Tikhonov} [J. Math. Anal. Appl. 347, No. ~2, 416--427 (2008; Zbl 1257.42009)] on the \(L^1-\)convergence of Fourier series.
reviewed by: Alexander Ulanovskii
Mathematics Subject Classification ID: 42A20
Mathematics Subject Classification ID: 42A32
zbMATH DE Number: 5857777
zbMATH Keywords: sine series
zbMATH Keywords: Fourier series
zbMATH Keywords: \(L^1\)-convergence
zbMATH Keywords: uniform convergence
zbMATH Keywords: monotonicity
zbMATH Keywords: general monotone
zbMATH Keywords: supremum bounded variation
author: Péter Kórus
MaRDI profile type: Publication
full work available at URL: s10474-010-9217-4
OpenAlex ID: W2032089841
cites work: On \(L_1\)-convergence of Fourier series
cites work: A generalization of the monotonicity condition and applications
cites work: Ultimate generalization to monotonicity for uniform convergence of trigonometric series
DOI: 10.1007/S10474-010-9217-4
description: scientific article; zbMATH DE number 5857777