Ring-like structures of frequency domains of wavelets
type: Item
instance of: scholarly article
title: Ring-like structures of frequency domains of wavelets
zbMATH Open document ID: 1196.42032
author: Naoki Saito
author: Zhihua Zhang
published in: Applied and Computational Harmonic Analysis
publication date: 2010-07-20T00:00:00Z
review text: The authors study orthonormal wavelets, i.e. sets \(\{\psi_\mu\}_{\mu=1}^{2^d-1}\) of functions in \(L^2(\mathbb R^d)\) such that their integer translations and dyadic dilations form an orthonormal basis, so the union \(\Omega\) of the supports of the Fourier transforms \(\hat {\psi_\mu}\) (the frequency domain) is a compact set. The set \(\Omega\) can then be written as \(\Omega = S \setminus S_*\) with \(0\in S_*\subset S\) and the authors give a characterization of these sets \(S\) and \(S_*\). As consequence of this characterization is that \(S\) and \(S_*\) cannot be balls. Reverse results on what sets can be the frequency domain of orthonormal wavelets. A number of examples are given as well.
reviewed by: Gustaf Gripenberg
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 5757857
zbMATH Keywords: orthonormal wavelet
zbMATH Keywords: support
zbMATH Keywords: regular set
MaRDI profile type: Publication
full work available at URL: j.acha.2009.08.003
OpenAlex ID: W2032463419
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DOI: 10.1016/J.ACHA.2009.08.003
description: scientific article; zbMATH DE number 5757857