Shift-modulation invariant spaces on LCA groups

TL;DR

This paper characterizes shift-modulation invariant spaces in $L^2(G)$ for locally compact abelian groups, extending known results from Euclidean spaces using fiberization and range functions.

Contribution

It generalizes the characterization of shift-modulation invariant spaces from Euclidean spaces to arbitrary LCA groups with uniform lattice subgroups.

Findings

Provides a fiberization technique for LCA groups.

Characterizes shift-modulation invariant spaces with range functions.

Extends previous Euclidean space results to a broader group context.

Abstract

A $(K,\Lambda)$ shift-modulation invariant space is a subspace of $L^2(G)$, that is invariant by translations along elements in $K$ and modulations by elements in $\Lambda$. Here $G$ is a locally compact abelian group, and $K$ and $\Lambda$ are closed subgroups of $G$ and the dual group $\hat G$, respectively. In this article we provide a characterization of shift-modulation invariant spaces in this general context when $K$ and $\Lambda$ are uniform lattices. This extends previous results known for $L^2(\R^d)$. We develop fiberization techniques and suitable range functions adapted to LCA groups needed to provide the desired characterization.