Duality of Gabor frames and Heisenberg modules

Abstract

Given a locally compact abelian group G and a closed subgroup Λ in G× G, Rieffel associated to Λ a Hilbert C * -module E, known as a Heisenberg module. He proved that E is an equivalence bimodule between the twisted group C * -algebra C * (Λ, c) and C * (Λ • , c), where Λ • denotes the adjoint subgroup of Λ. Our main goal is to study Heisenberg modules using tools from timefrequency analysis and pointing out that Heisenberg modules provide the natural setting of the duality theory of Gabor systems. More concretely, we show that the Feichtinger algebra S 0 (G) is an equivalence bimodule between the Banach subalgebras S 0 (Λ, c) and S 0 (Λ • , c) of C * (Λ, c) and C * (Λ • , c), respectively. Further, we prove that S 0 (G) is finitely generated and projective exactly for co-compact closed subgroups Λ. In this case the generators g 1 , . . . , g n of the left S 0 (Λ)-module S 0 (G) are the Gabor atoms of a multi-window Gabor frame for L 2 (G). We prove that this is equivalent to g 1 , . . . , g n being a Gabor super frame for the closed subspace generated by the Gabor system for Λ • . This duality principle is of independent interest and is also studied for infinitely many Gabor atoms. We also show that for any non-rational lattice Λ in R 2m with volume s(Λ) < 1 there exists a Gabor frame generated by a single atom in S 0 (R m ).

The Balian–Low theorem for locally compact abelian groups and vector bundles

“…We now proceed to review the construction by Rieffel in [54] of the modules that have been termed Heisenberg modules. We follow the approach by Jakobsen and Luef in [28] and use the Feichtinger algebra S 0 (G) instead of the Schwartz-Bruhat space S (G) from [54]. Although the Heisenberg module has a natural structure as an imprimitivity bimodule, only the left module structure will be important to us.…”

“…For instance, the Janssen representation of the Gabor frame operator can be seen as a consequence of the structure of Heisenberg modules. The duality theory of Gabor frames and Heisenberg modules was recently generalized to locally compact abelian groups by Luef and M. Jakobsen [28]. Moreover, an embedding of the Heisenberg module into L 2 (G) was demonstrated in [3].…”

On Gabor g-frames and Fourier series of operators

“…Remark 16. The fact that Gabor g-frames correspond to multi-window Gabor frames with countably many generators, suggests that the duality theory of Gabor g-frames (in the sense of Ron-Shen duality, see [33]) is covered by the approach in [39], where multi-window Gabor frames with countably many generators are considered.…”

Spectral invariance of $*$-representations of twisted convolution algebras with applications in Gabor analysis

“…We will do this by restating the problem in operator algebraic terms and then use Theorem 3.1. Exploring the interplay between Gabor analysis and operator algebras has gained much popularity in recent years [2,3,11,26,28,34,35]. The field of Gabor analysis has its origins in the seminal paper of Gabor [14], where he claimed that it is possible to obtain basis-like representations of functions in L 2 (R) in terms of the set {e 2πilx φ(x − k) : k, l ∈ Z}, where φ denotes a Gaussian.…”