Characterization and computation of canonical tight windows for Gabor frames
نویسنده
چکیده
Let (gna,mb)n,m∈Z be a Gabor frame for L (R) for given window g. We show that the window h0 = S− 1 2 g that generates the canonically associated tight Gabor frame minimizes ‖g − h‖ among all windows h generating a normalized tight Gabor frame. We present and prove versions of this result in the time domain, the frequency domain, the time-frequency domain, and the Zak transform domain, where in each domain the canonical h0 is expressed using functional calculus for Gabor frame operators. Furthermore, we derive a Wiener-Levy type theorem for rationally oversampled Gabor frames. Finally, a Newtontype method for a fast numerical calculation of h0 is presented. We analyze the convergence behavior of this method and demonstrate the efficiency of the proposed algorithm by some numerical examples. AMS Subject Classification: 42C15, 47A60, 94A11, 94A12.
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