Translation and dilation invariant subspaces of \(L^2(\mathbb{R})\) and multiresolution analyses
type: Item
instance of: scholarly article
title: Translation and dilation invariant subspaces of \(L^2(\mathbb{R})\) and multiresolution analyses
zbMATH Open document ID: 0926.42020
author: W. R. Madych
author: Rudolph Lorentz
author: Artur Sahakian
published in: Applied and Computational Harmonic Analysis
publication date: 1998-12-02T00:00:00Z
review text: If \(\tau=\{V_j\}_{j \in\mathbb{Z}}\) is a multiresolution analysis of \(L^2(\mathbb{R})\) then by definition \(\cup_{j\in \mathbb{Z}}V_j\) is dense in \(L^2(\mathbb{R})\). In this paper the authors consider multiresolution analysis which do not necessarily enjoy this property, and characterize the closure of \(\cup_{j \in\mathbb{Z}}V_j\). One consequence of their development is that in many instances the density hypothesis is redundant. Another consequence is the fact that every multiresolution analysis can be complemented by another one so that the orthogonal direct sum is dense in \(L^2(\mathbb{R})\).
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 1228529
zbMATH Keywords: multiresolution analysis
zbMATH Keywords: density hypothesis
reviewed by: Khalifa Trimèche
MaRDI profile type: Publication
full work available at URL: acha.1998.0235
OpenAlex ID: W2057110134
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DOI: 10.1006/ACHA.1998.0235
description: scientific article; zbMATH DE number 1228529