Characterizations of principal shift-invariant spaces.
type: Item
instance of: scholarly article
title: Characterizations of principal shift-invariant spaces.
zbMATH Open document ID: 1033.42041
author: Zhengchang Wu
published in: Applied Mathematics. Series B (English Edition)
publication date: 2004-03-22T00:00:00Z
review text: The paper considers shift-invariant spaces in \( L^1({\mathbb R})\). Given a compactly supported function \(\phi\), let \[ S(\phi ):= \biggl\{ f : f = \sum_{\alpha \in {\mathbb Z}} c(\alpha ) \phi(\cdot -\alpha ),\;c \in l({\mathbb Z}) \biggr\} \] be the principal shift-invariant space generated by \(\phi\). Then it has been shown by \textit{R.-Q. Jia} [Proc. Am. Math. Soc. 125, 785--793 (1997; Zbl 0860.41021)] that with \[ S_p(\phi) := \biggl\{ f : f = \sum_{\alpha \in {\mathbb Z} } c(\alpha) \phi (\cdot -\alpha),\;c \in l_p ({\mathbb Z}) \biggr\} \] we have for \( 1 < p < \infty \) \[ S_p( \phi ) = S(\phi ) \; \cap \; L_p({\mathbb R} ). \] For \(p = 1\), this assertion is not true. In this paper, shift-invariant spaces in \( L_1({\mathbb R} )\) are characterized and sufficient and necessary conditions are given, under which we still have \( S_1(\phi ) = S(\phi ) \, \cap \, L_1({\mathbb R})\). The investigations use special systems of difference equations and generators \(\phi \) being locally linearly independent over \([0, 1 ]\).
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 1836080
zbMATH Keywords: shift-invariant spaces
zbMATH Keywords: difference operators
zbMATH Keywords: linearly independent shifts
zbMATH Keywords: generator
reviewed by: Gerlind Plonka
MaRDI profile type: Publication
full work available at URL: s11766-002-0007-9
OpenAlex ID: W1967586218
cites work: The structure of finitely generated shift-invariant spaces in \(L_ 2(\mathbb{R}^ d)\)
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cites work: On linear independence for integer translates of a finite number of functions
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DOI: 10.1007/S11766-002-0007-9
description: scientific article; zbMATH DE number 1836080