Local approximation by certain spaces of exponential polynomials, approximation order of exponential box splines, and related interpolation problems

Local approximation order to smooth complex valued functions by a finite dimensional space K, spanned by certain products of exponentials by polynomials, is investigated. The results obtained, together with a suitable quasi-interpolation scheme, are used for the derivation of the approximation order attained by the linear span of translates of an exponential box spline.The analysis of a typical space K is based here on the identification of its dual with a certain space P of multivariate polynomials. This point of view allows us to solve a class of multivariate interpolation problems by the polynomials from P, with interpolation data characterized by the structure of K, and to construct bases of P corresponding to the interpolation problem.