Invariance of a Shift-Invariant Space in Several Variables

TL;DR

This paper investigates the invariance properties of shift-invariant spaces in multiple variables, providing conditions for invariance under subgroups and relating invariance to Fourier support, extending one-dimensional results.

Contribution

It extends the theory of shift-invariant spaces from one dimension to several variables, establishing conditions for subgroup invariance and Fourier support relations.

Findings

Characterization of invariance under closed subgroups in higher dimensions

Existence of spaces exactly invariant for each subgroup

Relation between invariance and Fourier transform support

Abstract

In this article we study invariance properties of shift-invariant spaces in higher dimensions. We state and prove several necessary and sufficient conditions for a shift-invariant space to be invariant under a given closed subgroup of $\R^d$, and prove the existence of shift-invariant spaces that are exactly invariant for each given subgroup. As an application we relate the extra invariance to the size of support of the Fourier transform of the generators of the shift-invariant space. This work extends recent results obtained for the case of one variable to several variables.