Real Algebraic and Analytic Geometry
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342. Murray Marshall:
Application of localization to the multivariate moment problem.

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Submission: 2012, May 1.

Abstract:
It is explained how the localization technique introduced by the author in an earlier paper leads to a useful reformulation of the multivariate moment problem in terms of extension of positive semidefinite linear functionals to positive semidefinite linear functionals on the localization of the polynomial ring at $p = \prod_{i=1}^n(1+x_i^2)$ or $p' = \prod_{i=1}^{n-1}(1+x_i^2)$. It is explained how this reformulation can be exploited to prove new results concerning existence and uniqueness of the measure $\mu$ and density of $\mathbb{C}[\underline{x}]$ in $\mathcal{L}^s(\mu)$ and, at the same time, to give new proofs of old results of Fuglede, Nussbaum, Petersen and Schm\"udgen, results which were proved previously using the theory of strongly commuting self-adjoint operators on Hilbert space.

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

Keywords and Phrases: positive definite, moments, sums of squares, Carleman condition.

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