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 funct...

Abstract Let and $H$?> be locally compact groups with fixed two-sided invariant Haar measures. A polyhomomorphism $G\rightarrowtail H$?> is a closed subgroup $R\subset G\times measure, whose marginals on are dominated by the measures . can regarded as multi-valued map sending points to sets equipped ‘uniform’ For two polyhomomorphisms ${G\rightarrowtail H}$?> $H\rightarrowtail K$?...