Homogeneous approximation property for continuous wavelet transforms
type: Item
instance of: scholarly article
title: Homogeneous approximation property for continuous wavelet transforms
zbMATH Open document ID: 1159.42021
author: Bei Liu
author: Wenchang Sun
published in: Journal of Approximation Theory
publication date: 2009-02-03T00:00:00Z
review text: The authors considers the continuous wavelet transform \[ \langle f,\tau(a,b)\psi\rangle = |a|^{-1/2} \int_{-\infty}^\infty f(x) \overline{\psi(a^{-1}(x-b))}\,dx. \] Under certain assumptions one can recover the function \(f\) by an inverse tranform and the question considered in this paper is to what extent this is possible to achieve in a uniform wayfor the time-scale shifts \(\tau(s,t)f(x)= |s|^{-1/2} f(s^{-1}(x-t))\), i.e., the homogeneous approximation property holds. It is shown that this is always the case in the \(L^2\)-sense but not for pointwise convergence and the authors give necessary and sufficient conditions, which depend both on the wavelet and the function \(f\), for the pointwise homogeneous approximation property to hold.
reviewed by: Gustaf Gripenberg
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 41A99
zbMATH DE Number: 5501961
zbMATH Keywords: continuous wavelet transform
zbMATH Keywords: homogeneous approximation property
MaRDI profile type: Publication
full work available at URL: j.jat.2008.03.010
OpenAlex ID: W2002011565
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DOI: 10.1016/J.JAT.2008.03.010
description: scientific article; zbMATH DE number 5501961