The Zak transform and the structure of spaces invariant by the action of an LCA group

TL;DR

This paper characterizes invariant subspaces of L^2 spaces under group actions using the Zak transform, providing a comprehensive framework for understanding their structure and applications to frames and Riesz sequences.

Contribution

It introduces a complete description of invariant subspaces via range functions and a generalized Zak transform, extending the analysis to translation-invariant spaces on LCA groups.

Findings

Characterization of invariant subspaces using range functions and Zak transform.

Criteria for frames and Riesz sequences generated by group actions.

Extension of results to translation-invariant subspaces on LCA groups.

Abstract

We study closed subspaces of $L^2(X)$, where $(X, \mu)$ is a $\sigma$-finite measure space, that are invariant under the unitary representation associated to a measurable action of a discrete countable LCA group $\Gamma$ on $X$. We provide a complete description for these spaces in terms of range functions and a suitable generalized Zak transform. As an application of our main result, we prove a characterization of frames and Riesz sequences in $L^2(X)$ generated by the action of the unitary representation under consideration on a countable set of functions in $L^2(X)$. Finally, closed subspaces of $L^2(G)$, for $G$ being an LCA group, that are invariant under translations by elements on a closed subgroup $\Gamma$ of $G$ are studied and characterized.