Duality of Gabor frames and Heisenberg modules

摘要

Given a locally compact abelian group G and a closed subgroup Lambda in G x (G) over cap, Rieffel associated to Lambda 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*(Lambda, c) and C*(Lambda degrees, (c) over bar), where Lambda degrees denotes the adjoint subgroup of Lambda. 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 S-0(G) is an equivalence bimodule between the Banach subalgebras S-0(Lambda, c) and S-0(Lambda degrees, (c) over bar) of C*(Lambda, c) and C*(Lambda degrees, (c) over bar), respectively. Further, we prove that S-0(G) is finitely generated and projective exactly for co-compact closed subgroups Lambda. In this case the generators g(1), . . ., g(n) of the left S-0(Lambda)-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 Lambda degrees. 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 Lambda in R-2m with volume s(Lambda) < 1 there exists a Gabor frame generated by a single atom in S-0(R-m).