This paper develops several aspects of shift invariant spaces on locally compact abelian groups. For a second countable locally compact abelian group G we prove a useful Hilbert space isomorphism, introduce range functions and give a characterization of shift invariant subspaces of L2(G) in terms of range functions. Utilizing these functions, we generalize characterizations of frames and Riesz bases generated by shifts of a countable set of generators from L2(Rn) to L2(G

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