Abstract
Let Delta be a closed, cocompact subgroup of G times widehat{G}, where G is a second countable, locally compact abelian group. Using localization of Hilbert C^*-modules, we show that the Heisenberg module mathcal {E}_{Delta }(G) over the twisted group C^*-algebra C^*(Delta ,c) due to Rieffel can be continuously and densely embedded into the Hilbert space L^2(G). This allows us to characterize a finite set of generators for mathcal {E}_{Delta }(G) as exactly the generators of multi-window (continuous) Gabor frames over Delta , a result which was previously known only for a dense subspace of mathcal {E}_{Delta }(G). We show that mathcal {E}_{Delta }(G) as a function space satisfies two properties that make it eligible for time-frequency analysis: Its elements satisfy the fundamental identity of Gabor analysis if Delta is a lattice, and their associated frame operators corresponding to Delta are bounded.
Highlights
Gabor analysis concerns sets of time-frequency shifts of functions
The author made the claim that one could obtain basis-like representations of functions in L2(R) in terms of the set {e2πilx φ(x − k) : k, l ∈ Z}, where φ denotes a Gaussian
We will allow continuous Gabor systems over any closed subgroup of the time-frequency plane G × G, which will be of the form (π(z)η)z∈
Summary
Gabor analysis concerns sets of time-frequency shifts of functions. The field has its roots in a paper by the electrical engineer and physicist Dennis Gabor [14]. The author made the claim that one could obtain basis-like representations of functions in L2(R) in terms of the set {e2πilx φ(x − k) : k, l ∈ Z}, where φ denotes a Gaussian. One of the central problems of the field remains understanding the
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