The Zak transform and the structure of spaces invariant by the action of an LCA group
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.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2014
- DOI:
- 10.48550/arXiv.1410.7250
- arXiv:
- arXiv:1410.7250
- Bibcode:
- 2014arXiv1410.7250B
- Keywords:
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- Mathematics - Functional Analysis;
- Mathematics - Group Theory