Relations between the support of a compactly supported function and the exponential-polynomials spanned by its integer translates

The interrelation between the shape of the support of a compactly supported function and the space of all exponential-polynomials spanned by its integer translates is examined. The results obtained are in terms of the behavior of these exponential-polynomials on certain finite subsets ofZ ^s , which are determined by the support of the given function. Several applications are discussed. Among these is the construction of quasi-interpolants of minimal support and the construction of a piecewise-polynomial whose integer translates span a polynomial space which is not scale-invariant. As to polynomial box splines, it is proved here that in many cases a polynomial box spline admits a certain optimality condition concerning the space of the total degree polynomials spanned by its integer translates: This space is maximal compared with the spaces corresponding to other functions with the same supportCommunicated by Klaus Höllig.