Stability of the shifts of a finite number of functions

Let phi(1), ..., phi(n), be compactly supported distributions in L-p(R-s) (0 < p less than or equal to infinity). We say that the shifts of phi(1), ..., phi(n) are L-p-stable if there exist two positive constants C-1 and C-2 such that, far arbitrary sequences a(1), .... a(n) is an element of l(p)(Z(s)), [GRAPHICS] In this paper we prove that the shifts of phi(1), ..., phi(n), are L-p-stable if and only if, for any xi is an element of R-s, the sequences (<(phi)over cap>(k)(xi + 2 beta pi))(beta) (is an element of) (Zs) (k = 1, ..., n) are linearly independent, where <(phi)over cap> denotes the Fourier transform of phi. This extends the previous results of Jia and Micchelli on a characterization of L-p-stability (1 less than or equal to p less than or equal to infinity) of the shifts of a finite number of compactly supported functions to the case 0 < p less than or equal to infinity. (C) 1998 Academic Press.