The moment problem with bounded density

Abstract

Let $μ$ be a given Borel measure on $\K\subseteq\R^n$ and let $y=(y_α)$, $α\in\N^n$, be a given sequence. We provide several conditions linking $y$ and the moment sequence $z=(z_α)$ of $μ$, for $y$ to be the moment sequence of a Borel measure $ν$ on $\K$ which is absolutely continuous with respect to $μ$ and such that its density is in $L_\infty(\K,μ)$. The conditions are necessary and sufficient if $\K$ is a compact basic semi-algebraic set, and sufficient if $\K\equiv\R^n$. Moreover, arbitrary finitely many of these conditions can be checked by solving either a semidefinite program or a linear program with a single variable