On estimates of partial sums of Fourier series of functions of bounded variation
type: Item
instance of: scholarly article
title: On estimates of partial sums of Fourier series of functions of bounded variation
zbMATH Open document ID: 0965.42002
published in: Russian Mathematics
publication date: 2000-12-07T00:00:00Z
review text: Let \(\{n_j\}\) be the sequence of all natural numbers \(k\) which satisfy the conditions \(3^p\leq k\leq 3^p+2^p,\) \(p=0,1,2,\dots .\) Then for any periodic function of bounded variation with the Fourier series \(a_0/2+\sum\limits_{m=1}^\infty (a_m\cos mx+b_m\sin mx)\) there holds \[ \sum_{j=1}^\infty \Biggl|\sum_{m=n_j}^ {n_{j+1}-1} (a_m\cos mx+b_m\sin mx) \Biggr|\leq \text{Const}\times\text{total variation of } f. \tag{1} \] This is the main result of the paper. Along with the assertion that there are functions for which (1) does not hold provided \(\lim_{j\to\infty} n_{j+1}/n_j=1,\) this is a certain refinement of an earlier result of the second author.
reviewed by: Elijah Liflyand
Mathematics Subject Classification ID: 42A16
Mathematics Subject Classification ID: 42A55
zbMATH DE Number: 1541567
zbMATH Keywords: Fourier series
zbMATH Keywords: bounded variation
zbMATH Keywords: partial sums
zbMATH Keywords: lacunary sequence
author: A. Yu Popov
author: S. A. Telyakovskiĭ
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1541567