Shift Invariant Spaces and Shift Preserving Operators on Locally Compact Abelian Groups

Abstract

We investigate shift invariant subspaces of L 2 (G), where G is a locally compact abelian group. We show that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame. For a second countable locally compact abelian group G we prove a useful Hilbert space isomorphism, introduce range functions and give a charac- terization of shift invariant subspaces of L 2 (G) in terms of range func- tions. Finally, we investigate shift preserving operators on locally com- pact abelian groups. We show that there is a one-to-one correspondence between shift preserving operators and range operators on L2(G )w here G is a locally compact abelian group.