**ABSTRACT**
### 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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