A DUAL BASIS FOR THE INTEGER TRANSLATES OF AN EXPONENTIAL BOX SPLINE

Exponential box splines are multivariate compactly supported functions on a regular mesh. Let phi be an exponential box spline associated with integer vectors. Then phi is piecewise in a space H spanned by exponential polynomials. In this paper we construct a dual basis for the integer translates of phi, when these translates are linearly independent. The dual basis is shown to be unique in a certain sense. Our construction is based on a study of the polynomial space F which consists of all polynomials p such that p(D)phi is a bounded function, where p(D) denotes the partial differential operator induced by p. It turns out that the linear space F is dual to H. Thus, as a by-product, we give a short proof for the formula of the dimension of H.