Positivity of Riesz Functionals and Solutions of Quadratic and Quartic Moment Problems

TL;DR

This paper uses positivity of Riesz functionals to characterize when truncated multivariate moment sequences have representing measures, solving several open cases of the truncated moment problem related to Hilbert's theorem.

Contribution

It establishes criteria based on Riesz functional positivity for the existence of representing measures, solving key open cases of the truncated moment problem.

Findings

Characterizes when truncated moment sequences admit representing measures.

Proves that strict K-positivity of Riesz functionals guarantees representing measures.

Solves open cases of the truncated moment problem in low degrees and dimensions.

Abstract

We employ positivity of Riesz functionals to establish representing measures (or approximate representing measures) for truncated multivariate moment sequences. For a truncated moment sequence $y$, we show that $y$ lies in the closure of truncated moment sequences admitting representing measures supported in a prescribed closed set $K \subseteq \re^n$ if and only if the associated Riesz functional $L_y$ is $K$-positive. For a determining set $K$, we prove that if $L_y$ is strictly $K$-positive, then $y$ admits a representing measure supported in $K$. As a consequence, we are able to solve the truncated $K$-moment problem of degree $k$ in the cases: (i) $(n,k)=(2,4)$ and $K=\re^2$; (ii) $n\geq 1$, $k=2$, and $K$ is defined by one quadratic equality or inequality. In particular, these results solve the truncated moment problem in the remaining open cases of Hilbert's theorem on sums of squares.