Shortest-support Multi-Spline Bases for Generalized Sampling

Abstract

Generalized sampling consists in the recovery of a function f , from the samples of the responses of a collection of linear shift-invariant systems to the input f . The reconstructed function is typically a member of a finitely generated integer-shift-invariant space that can reproduce polynomials up to a given degree M . While this property allows for an approximation power of order (M + 1), it comes with a tradeoff on the length of the support of the basis functions. Specifically, we prove that the sum of the length of the support of the generators is at least (M +1). Following this result, we introduce the notion of shortest basis of degree M , which is motivated by our desire to minimize the computational costs. We then demonstrate that any basis of shortest support generates a Riesz basis. Finally, we introduce a recursive algorithm to construct the shortest-support basis for any multi-spline space. It provides a generalization of both polynomial and Hermite B-splines. This framework paves the way for novel applications such as fast derivative sampling with arbitrarily high approximation power.