Absolute Riesz summability factors of Fourier series
type: Item
instance of: scholarly article
title: Absolute Riesz summability factors of Fourier series
zbMATH Open document ID: 0538.42003
author: Kôsi Kanno
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1984-01-01T00:00:00Z
review text: Let \(f\in L\) be a 2\(\pi\)-periodic function and let \(f(t)\sim \sum^{\infty}_{n=1}(a_ n\quad \cos nt+b_ n\quad \sin nt)\equiv \sum^{\infty}_{n=1}A_ n(t)\). Set \(\phi_ 0(t)=\phi(t)=((f(x+t)+f(x-t))/2\) and \(\phi_{\beta}(t)=\beta t^{- \beta}\int^{t}_{0}(t-u)^{\beta -1}\phi(u)du, \beta>0\). In this paper the author proves the following Theorem: Let \(0\leq \beta<1\), \(\delta\geq 0\), \(0<s+\beta<\delta +\beta<1\) and \(\alpha =(\delta - s)/(\delta +\beta)\) then \(t^{-\delta}\phi_{\beta}(t)\in BV[0,\pi]\) implies that \(\sum^{\infty}_{n=1}n^ sA_ n(x)\) is summable \(| R,\quad \exp \omega^{\alpha},\quad r|,\) with \(r>\delta +\beta\). Using this theorem, the author deduces known results due to \textit{K. Matsumoto} [Tôhoku Math. J., II. Ser. 8, 205-222 (1956; Zbl 0075.053)] and \textit{G. D. Dikshit} [Kyungpook Math. J. 17, 195-201 (1977; Zbl 0382.42005), J. Aust. Math. Soc., Ser. A 19, 97-102 (1975; Zbl 0293.42005), Indian J. Math. 18, 117-124 (1976; Zbl 0373.42006)].
Mathematics Subject Classification ID: 42A20
Mathematics Subject Classification ID: 40G05
Mathematics Subject Classification ID: 42A24
zbMATH DE Number: 3855694
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228900
OpenAlex ID: W2105238493
cites work: Q4161690
cites work: On absolute riesz summability of Fourier series
cites work: Q4150652
cites work: A sufficient condition for the absolute Riesz summability of a Fourier series
cites work: On absolute Cesàro summability of a series related to a Fourier series
cites work: On the Absolute Riesz Summability of Fourier Series and Allied Series
DOI: 10.2748/TMJ/1178228900
Recommended article: Q3779150
Recommended article: Q3709237
Recommended article: Q3736271
Recommended article: Q3803614
Recommended article: Q3032757
Recommended article: On absolute Riesz summability factors of infinite series and their application to Fourier series
Recommended article: On Riesz summability factors of Fourier series
Recommended article: Q5285288
Recommended article: Q4018888
Recommended article: Q5473255
description: scientific article; zbMATH DE number 3855694