  Let $(g_{nm})_{n,m\in Z}$ be a Gabor frame for $L_2(R)$ for given window $g$.
We show that the window $h^0=S^{-1/2} g$ 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 $h^0$
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 $\ho$ is
presented. We analyze the convergence behavior of this method and demonstrate
the efficiency of the proposed algorithm by some numerical examples.
