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arXiv · 1410.7250

The Zak transform and the structure of spaces invariant by the action of an LCA group

Davide Barbieri·Eugenio Hernández·Victoria Paternostro
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Abstract

We study closed subspaces of $L^2(X)$, where $(X, \mu)$ is a $\sigma$-finite measure space, that are invariant under the unitary representation associated to a measurable action of a discrete countable LCA group $\Gamma$ on $X$. We provide a complete description for these spaces in terms of range functions and a suitable generalized Zak transform. As an application of our main result, we prove a characterization of frames and Riesz sequences in $L^2(X)$ generated by the action of the unitary representation under consideration on a countable set of functions in $L^2(X)$. Finally, closed subspaces of $L^2(G)$, for $G$ being an LCA group, that are invariant under translations by elements on a closed subgroup $\Gamma$ of $G$ are studied and characterized.

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Davide Barbieri, Eugenio Hernández, Victoria Paternostro. 2014-10-27. The Zak transform and the structure of spaces invariant by the action of an LCA group. https://arxiv.org/abs/1410.7250

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