An algebraic perspective on multivariate tight wavelet frames
type: Item
instance of: scholarly article
title: An algebraic perspective on multivariate tight wavelet frames
zbMATH Open document ID: 1279.65146
author: Maria Charina
author: Mihai Putinar
author: Claus Scheiderer
author: Joachim Stöckler
published in: Constructive Approximation
publication date: 2013-11-11T00:00:00Z
review text: Using results of real algebraic geometry, the authors investigate the existence and construction of \(d\)-variate tight wavelet frames whose generators have at least one vanishing moment. The main results of the interesting paper are formulated as polynomial identities or polynomial inequalities. First, equivalent formulations of the unitary extension principle introduced by \textit{A. Ron} and \textit{Z. Shen} [J. Funct. Anal. 148, No. 2, 408--447 (1997; Zbl 0891.42018)] are given in terms of Hermitian sums of squares of certain nonnegative Laurent polynomials and in terms of semidefinite programming. These formulations lead to an affirmative answer to the question of the existence of tight wavelet frames in dimension \(d=2\). For \(d\geq 2\), the results provide efficient numerical methods for checking the existence of tight wavelet frames and for their construction. The authors show that for \(d=3\), the so-called sub-QMF condition is not sufficient for the existence of tight wavelet frames. Therefore, stronger sufficient conditions for the existence of tight wavelet frames are presented for \(d\geq 3\).
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 65T60
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 14P99
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 90C26
Mathematics Subject Classification ID: 90C22
zbMATH DE Number: 6225222
zbMATH Keywords: wavelet frame
zbMATH Keywords: multivariate tight wavelet frame
zbMATH Keywords: existence of tight wavelet frames
zbMATH Keywords: numerical construction of tight wavelet frames
zbMATH Keywords: real algebraic geometry
zbMATH Keywords: unitary extension principle
zbMATH Keywords: semidefinite programming
zbMATH Keywords: Laurent polynomials
MaRDI profile type: Publication
full work available at URL: s00365-013-9191-5
OpenAlex ID: W1981295076
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DOI: 10.1007/S00365-013-9191-5
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description: scientific article; zbMATH DE number 6225222