Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

On linear independence for integer translates of a finite number of functions

Rong-Qing Jiaa1 and Charles A. Micchellia2

a1 Department of Mathematics University of Alberta Edmonton, T6G 2G1, Canada

a2 IBM Research Division T. J. Watson Research Center Mathematical Sciences Department Yorktown Heights, NY 10598, USA

Abstract

We investigate linear independence of integer translates of a finite number of compactly supported functions in two cases. In the first case there are no restrictions on the coefficients that may occur in dependence relations. In the second case the coefficient sequences are restricted to be in some lp space (1 ≦ p ≦ ∞) and we are interested in bounding their lp-norms in terms of the Lp-norm of the linear combination of integer translates of the basis functions which uses these coefficients. In both cases we give necessary and sufficient conditions for linear independence of integer translates of the basis functions. Our characterization is based on a study of certain systems of linear partial difference and differential equations, which are of independent interest.

(Received January 03 1991)