Rational Noncommutative Tori and Gabor Frames
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A link between noncommutative geometry and time-frequency analysis is used to show that the TKNN equation violates existence results for Gabor frames with atoms in the Schwartz space. In particular, we use that the Schwartz space is an equivalence bimodule between smooth noncommutative tori associated to $\theta$ and $-1/\theta$. As a consequence, projections in smooth noncommutative tori correspond to tight Gabor frames generated by Schwartz functions. We calculate the trace and the Connes-Chern number of these projections and show that these provide counterexamples to the TKNN equation for rational smooth noncommutative tori.