Extra Invariance of Shift-Invariant Spaces on LCA Groups

TL;DR

This paper investigates the conditions under which shift-invariant spaces on locally compact abelian groups exhibit extra invariance properties, providing characterizations and constructing examples with precise invariance levels.

Contribution

It establishes necessary and sufficient conditions for H-invariant spaces to be M-invariant and constructs spaces with exact invariance properties in LCA groups.

Findings

Existence of H-invariant spaces exactly invariant under M

Characterization of invariance conditions in LCA groups

Support estimates for Fourier transforms of generators

Abstract

Let G be an LCA group and K a closed subgroup of G. A closed subspace of L^2(G) is called K-invariant if it is invariant under translations by elements of K. Assume now that H is a countable uniform lattice in G and M is any closed subgroup of G containing H. In this article we study necessary and sufficient conditions for an H-invariant space to be M-invariant. As a consequence of our results we prove that for each closed subgroup M of G containing the lattice H, there exists an H-invariant space S that is exactly M-invariant. That is, S is not invariant under any other subgroup M' containing M. We also obtain estimates on the support of the Fourier transform of the generators of the H-invariant space, related to its M-invariance.