In this article we extend the theory of shift-invariant spaces to the context of LCA groups. We introduce the notion of H -invariant space for a countable discrete subgroup H of an LCA group G, and show that the concept of range function and the techniques of fiberization are valid in this context. As a consequence of this generalization we prove characterizations of frames and Riesz bases of these spaces extending previous results, that were known for R d and the lattice Z d .
Abstract. We study closed subspaces of L 2 (X ), where (X , µ) is a σ-finite measure space, that are invariant under the unitary representation associated to a measurable action of a discrete countable LCA group Γ 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 Γ of G are studied and characterized. The results we obtain for this case are applicable to cases where those already proven in [5,7] are not.
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