Periodic wavelet frames
type: Item
instance of: scholarly article
title: Periodic wavelet frames
zbMATH Open document ID: 1063.42032
published in: Advances in Computational Mathematics
publication date: 2005-03-04T00:00:00Z
review text: The wavelet system associated to a function \(\psi\in L^2(\mathbb{R})\) consists of the functions \(\psi_{mn,}(x)=2^{m/2}\psi(2^mx-n), \;m,n\in Z\). Assume that the Fourier transform \(\hat{\psi}\) is twice differentiable, has support in \([-\pi, -\epsilon]\cup [\epsilon, \pi]\) for some \(\epsilon>0\), and satisfies that \(D(\gamma):=\sum_{m\in \mathbb{Z}} | \hat{\psi} (\gamma +m)| ^2>0\) for a.e. \(\gamma\). Define \(\tilde{\psi}\) by \(\hat{\tilde{\psi}}(\gamma)= \frac{\hat{\psi}(\gamma)}{D(\gamma)}\). Defining the periodization of a function \(f\) by \(f^{per}(t)=\sum_{k\in Z} f(t+k)\), it is proved that the functions in \(\Psi^{per}:= \{1, \psi_{m,n}^{per}, m\geq 0, n=0,1,\dots,2^m-1 \}\) and the corresponding functions in \(\tilde{\Psi}^{per}\) constitute dual frames for \(L^2(0,1)\). The optimal frame bounds for \(\Psi\) and \(\tilde{\Psi}\) are expressed in terms of the infimum and the supremum of the function \(D\).
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 2140538
zbMATH Keywords: periodic wavelet frames
zbMATH Keywords: periodization
author: Zhihua Zhang
MaRDI profile type: Publication
full work available at URL: s10444-004-3139-z
OpenAlex ID: W1973347232
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DOI: 10.1007/S10444-004-3139-Z
description: scientific article; zbMATH DE number 2140538