Shortest Multi-Spline Bases for Generalized Sampling
Alexis Goujon

Meeting • 24 November 2020

Abstract
Generalized sampling consists in the recovery of a function from the samples of its response to a collection of linear shift-invariant systems. The reconstructed function is typically chosen from a finitely generated shift-invariant space that can reproduce polynomials up to a given degree M. While this property guarantees an approximation power of order (M + 1), it comes with a tradeoff on the size of the support of the basis functions. Specifically, we prove that the sum of the supports of the generators is necessarily not smaller than (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 constitutes in 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.