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

Abstract

Given a multivariate compactly supported function $\phi $, linear projectors to the space $S(\phi )$ spanned by its integer translates are discussed here. These projectors are constructed with the aid of a dual basis for the integer translates of $\phi $, hence under the assumption that these translates are linearly independent. The main result shows that the linear functionals of the dual basis are local, hence making it possible to contruct local linear projectors onto $S(\phi )$. A scheme for the construction of such local projectors is then discussed for a general compactly supported function. In the second part of the paper these observations are applied 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.