Pointwise behavior of Fourier integrals of functions of bounded variation over \(\mathbb R\)
type: Item
instance of: scholarly article
title: Pointwise behavior of Fourier integrals of functions of bounded variation over \(\mathbb R\)
zbMATH Open document ID: 1065.42008
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-10-12T00:00:00Z
review text: Let \(S_T(f,x)=\int_{-T}^T \hat f(t) e^{ixt}dt,\) \(T>0,\) where \(\hat f\) is the Fourier transform of \(f.\) The author proves that if \(f\) is integrable and of bounded variation, written \(V(f)<\infty,\) on \(\mathbb R,\) and \(\sum_{j=m}^\infty u_j^{-1}\leq Au_m^{-1},\) \(m=1,2,\dots,\) for an increasing sequence \(\{u_j\}\) of positive numbers and a constant \(A,\) then \[ \sum_{j=1}^\infty \max_{u_{j-1}\leq \nu\leq u_j}| s_\nu(f,x)-s_{u_{j-1}} (f,x)| \leq{3A+4\over\pi}V(f), \quad x\in\mathbb R, \] with \(u_0=0.\) The series converges uniformly at every point of continuity of \(f\) or even on the whole closed interval of continuity. This implies various generalizations of results, known for Fourier series, to convergence of \(S_T(f,x)\) as well as a stronger version of the nonperiodic Parseval formula. The obtained main results are, in turn, generalizations and slight refinement of the periodic prototypes proved by S.A. Telyakovskij.
reviewed by: Elijah Liflyand
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 42A20
zbMATH DE Number: 2107205
zbMATH Keywords: Fourier integral
zbMATH Keywords: uniform convergence
zbMATH Keywords: Riemann-Stieltjes integral
zbMATH Keywords: bounded variation
zbMATH Keywords: Parseval formula
author: Ferenc Móricz
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.03.025
OpenAlex ID: W2093988234
cites work: Q5338374
cites work: Q5640160
cites work: Q4231102
cites work: On the uniform convergence of the Fourier series of functions of bounded variation
cites work: Q5769133
cites work: Q3254471
DOI: 10.1016/J.JMAA.2004.03.025
description: scientific article; zbMATH DE number 2107205