On the Convolution of a Box Spline with a Compactly Supported Distribution: Linear Independence for the Integer Translates

Abstract

The problem of linear independence of the integer translates of ?????where ? ?is a compactly supported distribution and ? is an exponential box spline, is considered in this paper. The main result relates the linear independence issue with the distribution of the zeros of the Fourier-Laplace transform ? of ? on certain linear manifolds associated with ?. The proof of our result makes an essential use of the necessary and sufficient condition derived in [11]. Several applications to specific situations are discussed. Particularly, it is shown that if the support of ? is small enough then linear independence is guaranteed provided that ? does not vanish at a certain finite set of critical points associated with ?. Also, the results here provoke a new proof of the linear independence condition for the translates of ? itself.