Local convergence of Fourier series with respect to periodized wavelets
type: Item
instance of: scholarly article
title: Local convergence of Fourier series with respect to periodized wavelets
zbMATH Open document ID: 0915.42021
author: K. Appert
published in: Journal of Approximation Theory
publication date: 1999-06-20T00:00:00Z
review text: The paper investigates the behavior of wavelet Fourier series of \(f\in L([0,1]^m)\) at Lebesgue and so-called strong Lebesgue points. Here \(x\) is said to be a Lebesgue point of \(f\) if \[ \lim_{h\rightarrow + 0}{1\over h^m}\int\limits_{[-h,h]^m}| f(x+t)-f(x) | dt=0. \] The wavelet Fourier basis of \(L_2([0,1]^m)\) is constructed by means of a tensor product of a one-dimensional multiresolution analysis of \(L^2([0,1])\) generated by a suitable orthogonal scaling function \(\varphi\). It is shown that the wavelet Fourier series converges at each strong Lebesgue point. Further, if \(\varphi\) and a corresponding orthogonal wavelet \(\psi\) satisfy a suitable decay condition then convergence at each Lebesgue point is found. The results can be transferred to the non-periodic case.
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 41A30
zbMATH DE Number: 1214022
zbMATH Keywords: wavelet Fourier series
zbMATH Keywords: periodic wavelets
zbMATH Keywords: strong Lebesgue points
zbMATH Keywords: multiresolution analysis
zbMATH Keywords: scaling function
reviewed by: Gerlind Plonka
MaRDI profile type: Publication
full work available at URL: jath.1998.3191
OpenAlex ID: W2018024502
cites work: Ten Lectures on Wavelets
cites work: Q3769379
cites work: Local convergence for wavelet expansions
cites work: On the strong derivatives of functions of intervals
cites work: Q4026702
DOI: 10.1006/JATH.1998.3191
description: scientific article; zbMATH DE number 1214022