Shortest-support Multi-Spline Bases for Generalized Sampling

TL;DR

This paper introduces shortest-support multi-spline bases for generalized sampling, optimizing the support length of basis functions to reduce computational costs while maintaining polynomial reproduction and approximation power.

Contribution

It establishes a lower bound on support length, defines shortest bases, proves their Riesz basis property, and provides a recursive construction algorithm for multi-spline spaces.

Findings

Shortest-support bases generate Riesz bases.

Recursive algorithm for constructing shortest-support bases.

Framework extends polynomial and Hermite B-splines.

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.