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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the real multidimensional rational $ K$-moment problem


Authors: Jaka Cimprič, Murray Marshall and Tim Netzer
Journal: Trans. Amer. Math. Soc. 363 (2011), 5773-5788
MSC (2010): Primary 44A60, 14P99
Published electronically: May 24, 2011
MathSciNet review: 2817409
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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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Additional Information

Jaka Cimprič
Affiliation: Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 21, SI-1000 Ljubljana, Slovenija
Email: cimpric@fmf.uni-lj.si

Murray Marshall
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5E6
Email: marshall@math.usask.ca

Tim Netzer
Affiliation: Fachbereich Mathematik und Informatik, Universität Leipzig, D-04009 Leipzig, Germany
Email: tim.netzer@math.uni-leipzig.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05225-6
PII: S 0002-9947(2011)05225-6
Keywords: Moment problem, positive polynomials, sums of squares
Received by editor(s): July 30, 2008
Received by editor(s) in revised form: October 9, 2009
Published electronically: May 24, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.