A note on the integer translates of a compactly supported distribution on \(\mathbb{R}\)
type: Item
instance of: scholarly article
title: A note on the integer translates of a compactly supported distribution on \(\mathbb{R}\)
zbMATH Open document ID: 0785.41006
DOI: 10.1007/BF01207192
author: Qiyu Sun
published in: Archiv der Mathematik
publication date: 1993-11-15T00:00:00Z
review text: Our objective in this note is to show a relation between global and local linear independence. To this, we introduce some definitions. Let \(\phi\) be a compactly supported distribution on \(R\). The integer translates of \(\phi\) are called globally linearly independent if the condition \(\sum_{k\in Z} c(k)\phi(x- k)=0\) on \(R\) implies \(c(k)=0\) for all \(k\in Z\). Let \(E^ k\) be the shift operator defined by \(E^ k\phi(x)= \phi(x+ k)\) in \(R\) for \(k\in Z\). \textit{A. Ben-Artzi} and \textit{A. Ron} [\((*)\) SIAM J. Math. Anal. 21, No. 6, 1550-1562 (1990; Zbl 0721.41021)] exhibit an equivalence between global linear independence and a very weak kind of local linear independence. They hoped that the theorem below (Claim 6.1 in \((*)\)) is true.
reviewed by: Qiyu Sun
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 446311
MaRDI profile type: Publication
cites work: On the Integer Translates of a Compactly Supported Function: Dual Bases and Linear Projectors
description: scientific article; zbMATH DE number 446311