Shortest Multi-spline Bases for Generalized Sampling
Alexis Goujon

Meeting • 03 August 2020

Abstract
Generalized sampling consists in recovering a function f from the samples of its response to N>=1 linear shift-invariant systems. Relevant reconstruction spaces include finitely generated shift-invariant spaces that are able to reproduce polynomials up to a given degree M. While this property guarantees an approximation power of order (M+1), it comes at a price: we prove that the sum of the size of the support of the generators is necessarily greater or equal than (M+1). When there is equality, the generating functions constitute a shortest support basis that is perfectly suited for applications since it minimizes the computation cost and, in addition, it necessarily forms a Riesz basis. Interestingly, for any multi-spline space $S_{n_1}+...+S_{n_N}$, a shortest-support basis can be constructed recursively, which generalises the well-known B-splines. These theoretical results pave the way for exciting applications, such as derivative sampling with arbitrarily high approximation power.