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

2014/10/27 by Barbieri, Davide, Hernández, Eugenio, Paternostro, Victoria · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1410.7250

Abstract

We study closed subspaces of L²(X), where (X, μ) is a σ-finite measure space, that are invariant under the unitary representation associated to a measurable action of a discrete countable LCA group Γ 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 L²(X) generated by the action of the unitary representation under consideration on a countable set of functions in L²(X). Finally, closed subspaces of L²(G), for G being an LCA group, that are invariant under translations by elements on a closed subgroup Γ of G are studied and characterized.

Cited by

• Ramanujan sums in signal recovery and uncertainty principle inequalities

Related

• Shift Invariant Spaces on LCA Groups
• Approximation by group invariant subspaces
• Multiplication-invariant operators and the classification of LCA group\n frames
• Extra invariance of principal shift invariant spaces and the Zak transform
• Extra Invariance of Shift-Invariant Spaces on LCA Groups