| Abstract: | Let (gnm)n,m ? Zst(gnm)_{n,minZst} be a Gabor frame for LtRLtR for given window gg. We show that the window ho = SQI gho=SQI g that generates the canonically associated tight Gabor frame minimizes ||g-h|||g-h| among all windows hh 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 hoho 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 hoho is presented. We analyze the convergence behavior of this method and demonstrate the efficiency of the proposed algorithm by some numerical examples. |
