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Local approximation order to smooth complex valued functions by a nite dimensional space H; 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 H is based here on the identication 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 H; and to construct bases of P corresponding to the interpolation problem.

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