On the real multidimensional rational $K$-moment problem
Jaka Cimprič, Murray A. Marshall, Tim Netzer · Transactions of the American Mathematical Society · 2011
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 $K_S$.