Shortest-support multi-spline bases for generalized sampling

Alexis Goujon, Shayan Aziznejad, Alireza Naderi, Michael Unser

Abstract: Generalized sampling consists in the recovery of a function f<math><mi is="true">f</mi></math>, from the samples of the responses of a collection of linear shift-invariant systems to the input f<math><mi is="true">f</mi></math>. 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<math><mi is="true">M</mi></math>. While this property allows for an approximation power of order (M+1)<math><mrow is="true"><mo is="true">(</mo><mi is="true">M</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mn is="true">1</mn><mo is="true">)</mo></mrow></math>, 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)<math><mrow is="true"><mo is="true">(</mo><mi is="true">M</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mn is="true">1</mn><mo is="true">)</mo></mrow></math>. Following this result, we introduce the notion of shortest basis of degree M<math><mi is="true">M</mi></math>, which is motivated by our desire to minimize 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.