Duality of Gabor frames and Heisenberg modules

Le résumé fourni par la source

Given a locally compact abelian group G and a closed subgroup \Lambda in G\times\hat{G} , Rieffel associated to \Lambda a Hilbert C^* -module \mathcal E , known as a Heisenberg module. He proved that \mathcal E is an equivalence bimodule between the twisted group C^* -algebra C^*(\Lambda, \mathsf{c}) and C^*(\Lambda^\circ,\bar{\mathsf{c}}) , where \Lambda^{\circ} 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 \mathbb S_O(G) is an equivalence bimodule between the Banach subalgebras \mathbb S_O(\Lambda,\mathsf{c}) and \mathbb S_O(\Lambda^\circ,\bar{\mathsf{c}}) of C^*(\Lambda,\mathsf{c}) and C^*(\Lambda^\circ,\bar{\mathsf{c}}) , respectively. Further, we prove that \mathbb S_O(G) is finitely generated and projective exactly for co-compact closed subgroups \Lambda . In this case the generators g_1,\ldots,g_n of the left \mathbb S_O(\Lambda) -module \mathbb S_O(G) are the Gabor atoms of a multi-window Gabor frame for L^2(G) . We prove that this is equivalent to g_1,\ldots,g_n being a Gabor super frame for the closed subspace generated by the Gabor system for \Lambda^\circ . 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 \mathbb R^{2m} with volume s(\Lambda) < 1 there exists a Gabor frame generated by a single atom in \Lambda^\circ(\mathbb R^m) .

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