Constructive realization of dual systems for generators of multi-window spline-type spaces

Multi-window spline-type spaces arise naturally in many areas. Among others they have been used as model spaces in the theory of irregular sampling. This class of shift-invariant spaces is characterized by possessing a Riesz basis   which consists of a set of translates along some lattice ΛΛ of a finite family of atoms. Part of their usefulness relies on the explicit knowledge of the structure of the projection operator on such a space using the existence of a finite family of dual atoms. The main goal of this paper is to address the problems arising from the discrepancy between a constructive description and an implementable approximate realization of such concepts. Using function space concepts (e.g. Wiener amalgam spaces) we describe how approximate dual atoms can be computed for any given degree of precision.As an application of our result we describe the best approximation of Hilbert–Schmidt operators by generalized Gabor multipliers, using smooth analysis and synthesis windows. The Kohn–Nirenberg symbols of the rank-one operators formed from analysis and synthesis windows satisfy our general assumptions. Applications to irregular sampling are given elsewhere.