  We consider the problem of approximating the unknown density $u\in
L^2(\Omega,\lambda)$ of a measure $\mu$ on $\Omega\subset\R^n$, absolutely
continuous with respect to some given reference measure $\lambda$, from the
only knowledge of finitely many moments of $\mu$. Given $d\in\N$ and moments of
order $d$, we provide a polynomial $p_d$ which minimizes the mean square error
$\int (u-p)^2d\lambda$ over all polynomials $p$ of degree at most $d$. If there
is no additional requirement, $p_d$ is obtained as solution of a linear system.
In addition, if $p_d$ is expressed in the basis of polynomials that are
orthonormal with respect to $\lambda$, its vector of coefficients is just the
vector of given moments and no computation is needed. Moreover $p_d\to u$ in
$L^2(\Omega,\lambda)$ as $d\to\infty$. In general nonnegativity of $p_d$ is not
guaranteed even though $u$ is nonnegative. However, with this additional
nonnegativity requirement one obtains analogous results but computing
$p_d\geq0$ that minimizes $\int (u-p)^2d\lambda$ now requires solving an
appropriate semidefinite program. We have tested the approach on some
applications arising from the reconstruction of geometrical objects and the
approximation of solutions of nonlinear differential equations. In all cases
our results are significantly better than those obtained with the maximum
entropy technique for estimating $u$.
