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