Shift-modulation invariant spaces on LCA groups

A $(K,Λ)$ 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 $Λ$. Here $G$ is a locally compact abelian group, and $K$ and $Λ$ 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 $Λ$ 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.

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