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

By Davide Barbieri, E. Hern'andez, Victoria Paternostro. 2014

62 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Signal Processing

View Paper: https://www.semanticscholar.org/paper/b8831c11b784b584915136f55d7286af6db642fe

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Citations in the corpus

Iterative actions of normal operators (2016, background): “For us, the motivation to study the iterative actions of normal operators comes from sampling theory and related topics [2, 9, 21, 29, 31, 5, 6, 18, 37, 38, 16, 15].”
Subspaces of L2(G) invariant under translation by an abelian subgroup (2014, unknown): “utility of Tcomes from the intertwining relation T(T kf)(x) = e2ˇikxTf(x) for all k2Zn; 1While putting the nishing touches on this paper, the author learned that Barbieri, Hernandez, and Paternostro [8] simultaneously and independently developed a classication of H-TI spaces using an identical notio”