Determination of the jumps of a bounded function by its Fourier series
type: Item
instance of: scholarly article
title: Determination of the jumps of a bounded function by its Fourier series
zbMATH Open document ID: 0902.42001
author: George Kvernadze
published in: Journal of Approximation Theory
publication date: 1998-12-10T00:00:00Z
review text: It is well known that the jumps of a function of bounded variation can be determined by means of the partial sums of its differentiated Fourier series as follows: \[ \lim_{n\to\infty} {(S_n(f;\theta))'\over n}= {1\over\pi} (f(\theta+)- f(\theta-)),\tag{\(*\)} \] where \(f\) is a \(2\pi\)-periodic function of bounded variation on \([-\pi,\pi]\), in sign: \(f\in V\). \textit{B. I. Golubov} [Mat. Zametki 12,19-28 (1972; Zbl 0256.42005)] generalized \((*)\) for classes \(V_p\) with \(p\geq 1\) and higher derivatives of the Fourier partial sums. Now, the present author extends \((*)\) to even larger classes of functions, such as \(V_\phi\), \(\Lambda\text{BV}\) (introduced by D. Waterman), and \(V[v]\) (introduced by Z. A. Čanturija), under some additional conditions on the generalized variation. It is also shown that these additional conditions are the best possible in certain sense.
Mathematics Subject Classification ID: 42A16
zbMATH DE Number: 1146507
zbMATH Keywords: jumps of a function of bounded variation
zbMATH Keywords: derivatives of the Fourier partial sums
zbMATH Keywords: generalized variation
reviewed by: Ferenc Móricz
MaRDI profile type: Publication
full work available at URL: jath.1997.3125
OpenAlex ID: W2030382489
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DOI: 10.1006/JATH.1997.3125
description: scientific article; zbMATH DE number 1146507