  In this paper we introduce the theory of derivatives of moments and (moment)
functionals to represent moment functionals by Gaussian mixtures,
characteristic functions of polytopes, and simple functions of polytopes. We
study, among other measures, Gaussian mixtures, their reconstruction from
moments and especially the number of Gaussians needed to represent moment
functionals. We find that there are moment functionals
$L:\mathbb{R}[x_1,\dots,x_n]_{\leq 2d}\to\mathbb{R}$ which can be represented
by a sum of $\binom{n+2d}{n} - n\cdot \binom{n+d}{n} + \binom{n}{2}$ Gaussians
but not less. Hence, for any $d\in\mathbb{N}$ and $\varepsilon>0$ we find an
$n\in\mathbb{N}$ such that $L$ can be represented by a sum of
$(1-\varepsilon)\cdot\binom{n+2d}{n}$ Gaussians but not less. An upper bound is
$\binom{n+2d}{n}-1$.
