On the series of absolute values of blocks of trigonometric series
type: Item
instance of: scholarly article
title: On the series of absolute values of blocks of trigonometric series
zbMATH Open document ID: 1306.42005
DOI: 10.1134/S0081543814010179
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2014-12-17T00:00:00Z
review text: Let \(\Lambda\) be a strictly increasing sequence of natural numbers, \(1=n_{1}<n_{2}< ... \). To the series \(\sum_{k=0}^{\infty}a_{k}\cos(k x)\) and \(\sum_{k=0}^{\infty}b_{k}\sin(k x)\), with \(a_{k}, b_{k}\in \mathbb{R}\), \(k\in \mathbb{N}\cup \{0\},\) and \(\lim_{k\to \infty}a_{k}=\lim_{k\to \infty}b_{k}=0\), one associates the series \[ F_{\Lambda}(x)=\sum_{j=1}^{\infty}\left |\sum_{k=n_{j}}^{n_{j+1}-1}a_{k}\cos(k x)\right |, \, \, G_{\Lambda}(x)=\sum_{j=1}^{\infty}\left |\sum_{k=n_{j}}^{n_{j+1}-1}b_{k}\sin(k x)\right |, \] where evidently the values of \(F_{\Lambda}(x)\) and \(G_{\Lambda}(x)\) may be \(+\infty\). The main results of the paper can be stated as follows. { Theorem.} Let \(\gamma\in (0, 1)\), \(a_{k}, b_{k}\in \mathbb{R}\), \(k\in \mathbb{N}\cup \{0\},\) with \(\lim_{k\to \infty}a_{k}=\lim_{k\to \infty}b_{k}=0\). (i) If the series \(\sum_{j=1}^{\infty}\sum_{k=n_{j}}^{n_{j+1}-1}|a_{k}-a_{k+1}|(k-n_{j}+1)^{\gamma}\) converges, then the integral \(\int_{0}^{\pi}\frac{1}{x^{\gamma}}\cdot F_{\Lambda}(x)dx \) converges if and only if the series \(\sum_{j=1}^{\infty}|a_{n_{j+1}}|(n_{j+1}-n_{j})^{\gamma}\) converges. (ii) If the series \(\sum_{j=1}^{\infty}\sum_{k=n_{j}}^{n_{j+1}-1}|a_{k}-a_{k+1}|\log(k-n_{j}+2)\) converges, then \(F_{\Lambda}\) is Lebesgue integrable on \([0, \pi]\) if and only if the series \(\sum_{j=1}^{\infty}|a_{n_{j+1}}\log(n_{j+1}-n_{j}+1)\) converges. (iii) If the series \(\sum_{j=1}^{\infty}\sum_{k=n_{j}}^{n_{j+1}-1}|b_{k}-b_{k+1}|(k-n_{j}+1)^{\gamma}\) converges, then the integral \(\int_{0}^{\pi}\frac{1}{x^{\gamma}}\cdot G_{\Lambda}(x)dx \) converges if and only if the series \(\sum_{j=1}^{\infty}|b_{n_{j+1}}|(n_{j+1}-n_{j})^{\gamma}\) converges.
Mathematics Subject Classification ID: 42A24
Mathematics Subject Classification ID: 42A32
zbMATH DE Number: 6381017
zbMATH Keywords: Fourier series
zbMATH Keywords: summability
zbMATH Keywords: blocks of trigonometric series
author: S. A. Telyakovskiĭ
reviewed by: Sorin G. Gal
MaRDI profile type: Publication
cites work: Q5469274
cites work: Some properties of the sum of the moduli of the terms of a grouped trigonometric series
cites work: On the properties of blocks of terms of the series \(\sum\frac{1}{k}\sin k\,x\)
cites work: Q4979633
description: scientific article; zbMATH DE number 6381017