Duality of Gabor frames and Heisenberg modules

  • Mads S. Jakobsen

    Norwegian University of Science and Technology (NTNU), Trondheim, Norway
  • Franz Luef

    Norwegian University of Science and Technology (NTNU), Trondheim, Norway
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Abstract

Given a locally compact abelian group and a closed subgroup in , Rieffel associated to a Hilbert -module , known as a Heisenberg module. He proved that is an equivalence bimodule between the twisted group -algebra and , where denotes the adjoint subgroup of . Our main goal is to study Heisenberg modules using tools from time-frequency 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 is an equivalence bimodule between the Banach subalgebras and of and , respectively. Further, we prove that is finitely generated and projective exactly for co-compact closed subgroups . In this case the generators of the left -module are the Gabor atoms of a multi-window Gabor frame for . We prove that this is equivalent to being a Gabor super frame for the closed subspace generated by the Gabor system for . 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 in with volume there exists a Gabor frame generated by a single atom in .

Cite this article

Mads S. Jakobsen, Franz Luef, Duality of Gabor frames and Heisenberg modules. J. Noncommut. Geom. 14 (2020), no. 4, pp. 1445–1500

DOI 10.4171/JNCG/413