On Linear Dependence Relations for Integer Translates of Compactly Supported Distributions

Abstract

A necessary and sufficient condition for the dimension of the space of dependence relations for (multi‐) integer translates of an arbitrary compactly supported distribution in terms of zeros of its Fourier transform is given. We apply this result to obtain necessary and sufficient conditions on an integer matrix X so that the space of dependence relations for the corresponding cube spline C(·|X) is finite dimensional. We are able to describe the explicit form of all dependence relations. Copyright © 1991 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim