Affine systems in \(L_ 2(\mathbb{R}^d)\): The analysis of the analysis operator
type: Item
instance of: scholarly article
title: Affine systems in \(L_ 2(\mathbb{R}^d)\): The analysis of the analysis operator
zbMATH Open document ID: 0891.42018
author: A. Ron
author: Zuowei Shen
published in: Journal of Functional Analysis
publication date: 1997-12-14T00:00:00Z
review text: The generation of wavelets from a multiresolution analysis of \(L_2({\mathbb{R}}^d)\) is usually restricted to (bi)orthogonal systems which are not redundant. However, when switching to frames, one runs into difficulties. This paper gives a framework to tackle these problems. Suppose we work on lattices in \({\mathbb{R}}^d\). An affine system \(X\) consists of the orbits generated by a discrete analog of the affine group (``dilations'' and ``translations'') applied to a finite set of functions (``mother functions''). The problem of deciding whether such a system is a (tight) frame is approached via an in depth discussion of the analysis operator. That is the operator \(T_X^*\) which maps \(f\in L_2({\mathbb{R}}^d)\) onto the sequence of coefficients \((\langle f,x\rangle)_{x\in X}\in\ell_2(X)\) (it is the adjoint of the synthesis operator). The authors finally arrive at the construction of a tight frame via a multiresolution analysis. This paper is a continuation of the authors' previous papers on this subject: [Can. J. Math. 47, No. 5, 1051-1094 (1995; Zbl 0838.42016) and ``Weyl-Heisenberg frames and Riesz bases in \(L_2({\mathbb{R}}^d)\)'', Duke Math. J. (in press)].
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 42C30
zbMATH DE Number: 1069301
zbMATH Keywords: wavelets
zbMATH Keywords: tight frames
zbMATH Keywords: multiresolution analysis
reviewed by: Adhemar Bultheel
MaRDI profile type: Publication
full work available at URL: jfan.1996.3079
OpenAlex ID: W1983772857
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DOI: 10.1006/JFAN.1996.3079
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description: scientific article; zbMATH DE number 1069301