Lattice invariant subspaces and sampling

We introduce a sampling theory for cyclic lattice invariant spaces in the context of locally compact Abelian groups. The key element of the theory is a new periodization condition which, for L a lattice in a group G, characterizes the L cyclic invariant subspaces V⊆L2(G) which allow sampling. When L is contained in a larger lattice L′, we characterize the collection of L-invariant subspaces of L2(G) which are also L′-invariant. We use this to show that, if V is cyclic with respect to both L and L′, then V admits sampling with respect to L if and only if V admits sampling with respect to L′.