A characterization of subspaces generated by the action of a discrete LCA group on a measure space

PATERNOSTRO VICTORIA

Conferencia; Strobl14 Modern Time-Frequency Analysis; 2014

Given $\Gamma$ a discrete locally compact abelian (LCA) group acting in a measure space $(X,\mu)$, we study closed subspaces of $L^2(X)$ thata re invariant under the action of $\Gamma$. We give a characterization of these spaces in terms of a suitable Zak transform and range functions. We use this characterization to obtain necessary and sufficient conditions for a system generated by the action of $\Gamma$ on a set of functions in $L^2(X)$ to be a frame for the space it spans. We then connect our results with those previously proven for the case when a uniform lattice $H$ on a group $G$ acts by translations on $L^2(G)$. We provode examples and discuss some possible extensions to the non-abelian setting. There results are part of a joint work with Davide Barbieri and Eugenio Hernández.