Sampling formulas involving differences in shift-invariant subspaces: a unified approach

TL;DR

This paper develops a unified approach to sampling formulas involving differences in shift-invariant subspaces, enabling stable function recovery and illustrating various scenarios including two-dimensional cases.

Contribution

It introduces a novel method to derive sampling formulas using differences in shift-invariant subspaces, with a focus on stability and a unified framework.

Findings

Derived explicit formulas for functions in shift-invariant subspaces.

Illustrated the approach with examples, including two-dimensional cases.

Showed the stability of the recovery process.

Abstract

Successive differences on a sequence of data help to discover some smoothness features of this data. This was one of the main reasons for rewriting the classical interpolation formula in terms of such data differences. The aim of this paper is to mimic them to a sequence of regular samples of a function in a shift-invariant subspace allowing its stable recovery. A suitable expression for the functions in the shift-invariant subspace by means of an isomorphism with the $L^2(0,1)$ space is the key to identify the simple pattern followed by the dual Riesz bases involved in the derived formulas. The paper contains examples illustrating different non-exhaustive situations including also the two-dimensional case.