The K-moment problem with densities

Given a compact basic semi-algebraic set ${\mathbf{K}} \subset {\mathbb{R}}^n$ , a rational fraction $f:{\mathbb{R}}^n\to{\mathbb{R}}$ , and a sequence of scalars y = (y _α), we investigate when $y_\alpha =\int_{\mathbf{K}} x^\alpha f\,d\mu$ holds for all $\alpha\in{\mathbb{N}}^n$ , i.e., when y is the moment sequence of some measure fdμ, supported on K. This yields a set of (convex) linear matrix inequalities (LMI). We also use semidefinite programming to develop a converging approximation scheme to evaluate the integral ∫ fdμ when the moments of μ are known and f is a given rational fraction. Numerical expreriments are also provided. We finally provide (again LMI) conditions on the moments of two measures $\nu,\mu$ with support contained in K, to have $d\nu=f d\mu$ for some rational fraction f.