  In this short note we study how well a Gaussian distribution can be
approximated by distributions supported on $[-a,a]$. Perhaps, the natural
conjecture is that for large $a$ the almost optimal choice is given by
truncating the Gaussian to $[-a,a]$. Indeed, such approximation achieves the
optimal rate of $e^{-\Theta(a^2)}$ in terms of the $L_\infty$-distance between
characteristic functions. However, if we consider the $L_\infty$-distance
between Laplace transforms on a complex disk, the optimal rate is
$e^{-\Theta(a^2 \log a)}$, while truncation still only attains
$e^{-\Theta(a^2)}$. The optimal rate can be attained by the Gauss-Hermite
quadrature. As corollary, we also construct a ``super-flat'' Gaussian mixture
of $\Theta(a^2)$ components with means in $[-a,a]$ and whose density has all
derivatives bounded by $e^{-\Omega(a^2 \log(a))}$ in the $O(1)$-neighborhood of
the origin.
