A note on the integer translates of a compactly supported distribution on \(\mathbb{R}\)
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.
scientific article; zbMATH DE number 446311