Real Algebraic and Analytic Geometry |

On the real multidimensional rational $K$-moment problem.

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Submission: 2008, July 12.

*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 $E=D^{-1}\RR[\underline{X}]$ be a localization of
the real polynomial algebra, and $T_S^E$ 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 $K_S$.*

Mathematics Subject Classification (2000): 44A60, 14P99.

Keywords and Phrases: moment problem, positive polynomials, sums of squares.

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