Borel measures with a density on a compact semi-algebraic set

TL;DR

This paper characterizes when a sequence corresponds to a Borel measure with a density in various Lp spaces on a compact semi-algebraic set, using semidefinite programming hierarchies for detection.

Contribution

It provides necessary and sufficient conditions for the existence of absolutely continuous measures with densities in Lp spaces, without initial bounds, using a hierarchy of semidefinite programs.

Findings

Characterization of measures with densities in Lp spaces on semi-algebraic sets.

Hierarchy of semidefinite programs can detect nonexistence of such densities.

No a priori bounds needed for the main characterization.

Abstract

Let $K\subset R^n$ be a compact basic semi-algebraic set. We provide a necessary and sufficient condition (with no a priori bounding parameter) for a real sequence $y=(y_\alpha)$, $\alpha\in N^n$, to have a finite representing Borel measure absolutely continuous w.r.t. the Lebesgue measure on $K$, and with a density in $\cap_{p=1}^\infty L_p(K)$. With an additional condition involving a bounding parameter, the condition is necessary and sufficient for existence of a density in $L_\infty(K)$. Moreover, nonexistence of such a density can be detected by solving finitely many of a hierarchy of semidefinite programs. In particular, if the semidefinite program at step $d$ of the hierarchy has no solution then the sequence cannot have a representing measure on $K$ with a density in $L_p(K)$ for any $p\geq 2d$.