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arXiv · 0908.3230

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

Lawrence Fialkow·Jiawang Nie
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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.

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Lawrence Fialkow, Jiawang Nie. 2009-09-16. Positivity of Riesz Functionals and Solutions of Quadratic and Quartic Moment Problems. https://arxiv.org/abs/0908.3230

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