Lebesgue decomposition in action via semidefinite relaxations

Abstract

Given all (finite) moments of two measures $μ$ and $λ$ on $\R^n$, we provide a numerical scheme to obtain the Lebesgue decomposition $μ=ν+ψ$ with $ν\llλ$ and $ψ\perpλ$. When$ν$ has a density in $L\_\infty(λ)$ then we obtain two sequences of finite moments vectorsof increasing size (the number of moments) which converge to the moments of $ν$ and $ψ$ respectively, as the number of moments increases. Importantly, {\it no} à priori knowledge on the supports of $μ, ν$ and $ψ$ is required.