On linear independence for integer translates of a finite number of functions
Rong-Qing Jia, Charles A. Micchelli · Proceedings of the Edinburgh Mathematical Society · 1993
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 somelpspace (1 ≦p≦ ∞) and we are interested in bounding theirlp-norms in terms of theLp-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.