On the Integer Translates of a Compactly Supported Function: Dual Bases and Linear Projectors

Abstract

Given a multivariate compactly supported function �, we discuss here linear projectors to the space S (�) spanned by its integer translates. These projectors are constructed with the aid of a dual basis for the integer translates of �, hence under the assumption that these translates are linearly independent. Our main result shows that the linear functionals of the dual basis are local, hence makes it possible to construct local linear projectors onto S(�). We then discuss, for a general compact.ly supported function, a scheme for the construction of such local projectors. In the second part of the paper we apply these observations to piecewise-polynomials and piecewise-exponentials to obtain a necessary and sufficient condition for a quasi-interpolant to be a projector. The results of that part extend and refine recent constructions of dual bases and linear projectors for polynomial and exponential box splines.