On the real multidimensional rational 𝐾-moment problem

Cimprič, Jaka; Marshall, Murray; Netzer, Tim

doi:10.1090/S0002-9947-2011-05225-6

On the real multidimensional rational $K$-moment problem
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by Jaka Cimprič, Murray Marshall and Tim Netzer

Trans. Amer. Math. Soc. 363 (2011), 5773-5788

DOI: https://doi.org/10.1090/S0002-9947-2011-05225-6

Published electronically: May 24, 2011

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Abstract:

We present a solution to the real multidimensional rational $K$-moment problem, where $K$ is defined by finitely many polynomial inequalities. More precisely, let $S$ be a finite set of real polynomials in $\underline {X}=(X_1,\ldots ,X_n)$ such that the corresponding basic closed semialgebraic set $K_S$ is nonempty. Let $E=D^{-1}\mathbb {R}[\underline {X}]$ be a localization of the real polynomial algebra and let $T_S^E$ be the preordering on $E$ generated by $S$. We show that every linear functional $L$ on $E$ such that $L(T_S^E) \ge 0$ is represented by a positive measure $\mu$ on a certain subset of $K_S$, provided $D$ contains an element that grows fast enough on $K_S$.

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