Characterization and computation of canonical tight windows for Gabor frames

A.J.E.M. Janssen and Thomas Strohmer

Let $g_{n,m}_{n,m in Z}$ be a Gabor frame for $L^2(R)$ for given window $g$. We show that the window $h^o= S^{-1/2} g$ 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 $h^o$ 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 $h^o$ is presented. We analyze the convergence behavior of this method and demonstrate the efficiency of the proposed algorithm by some numerical examples.

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