Characterization and computation of canonical tight windows for Gabor frames

Let (gna,mb)n,m¿Z be a Gabor frame for L2 (R) for given window g. We show that the window h0 = S-1/2g 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 Newton-type 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