Stability analysis of biorthogonal multiwavelets whose duals are not in \(L_2\) and its application to local semiorthogonal lifting
type: Item
instance of: scholarly article
title: Stability analysis of biorthogonal multiwavelets whose duals are not in \(L_2\) and its application to local semiorthogonal lifting
zbMATH Open document ID: 1147.65113
author: Jan Maes
author: Adhemar Bultheel
published in: Applied Numerical Mathematics
publication date: 2008-07-11T00:00:00Z
full work available at URL: jfa05
review text: The authors investigate the Riesz stability of compactly supported (multi-)wavelet bases that are constructed by a lifting scheme on regularly refined meshes of arbitrary topological type. The authors are interested in the Riesz stability of the two-step lifted wavelet transform consisting of one prediction step and one update step. The prediction step is based on subdivision rules. The update step can be described as local semiorthogonalization. In order to compute the exact range of Sobolev exponents \(s\) for which the wavelets form a Riesz basis of the Sobolev space \(H^s({\mathbb R}^d)\), one needs to determine the smoothness of the dual wavelet system. It might occur that the duals do not exist in \(L_2({\mathbb R}^d)\). By Fourier techniques, the range of corresponding Sobolev exponents is estimated. Examples of uni- and bivariate wavelet systems are presented. These examples show that the lifted wavelets are a Riesz basis for a larger range of Sobolev exponents than the corresponding non-updated hierarchical bases but in general they do not form a Riesz basis in \(L_2({\mathbb R}^d)\).
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 65T60
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 5300753
zbMATH Keywords: biorthogonal wavelets
zbMATH Keywords: compactly supported wavelet basis
zbMATH Keywords: multiwavelets
zbMATH Keywords: multivariate wavelets
zbMATH Keywords: Riesz stability
zbMATH Keywords: lifting scheme
zbMATH Keywords: range of Sobolev exponents
zbMATH Keywords: subdivision
zbMATH Keywords: local semiorthogonal lifting
MaRDI profile type: Publication
OpenAlex ID: W2047364218
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DOI: 10.1016/J.APNUM.2007.06.002
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description: scientific article; zbMATH DE number 5300753