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 .

Let Y = {f (i), Af (i), . . . , A l i f (i) : i ∈ Ω}, where A is a bounded operator on 2 (I). The problem under consideration is to find necessary and sufficient conditions on A, Ω, {li : i ∈ Ω} in order to recover any f ∈ 2 (I) from the measurements Y . This is the so called dynamical sampling problem in which we seek to recover a function f by combining coarse samples of f and its futures states A l f . We completely solve this problem in finite dimensional spaces, and for a large class of self adjoint operators in infinite dimensional spaces. In the latter case, the Müntz-Szász Theorem combined with the Kadison-Singer/Feichtinger Theorem allows us to show that Y can never be a Riesz basis when Ω is finite. We can also show that, when Ω is finite, Y = {f (i), Af (i), . . . , A l i f (i) : i ∈ Ω} is not a frame except for some very special cases. The existence of these special cases is derived from Carleson's Theorem for interpolating sequences in the Hardy space H 2 (D).