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

 

Solution of the truncated moment problem with variety $ y = x^{3}$


Author: Lawrence A. Fialkow
Journal: Trans. Amer. Math. Soc. 363 (2011), 3133-3165
MSC (2000): Primary 47A57, 47A20, 44A60; Secondary 15A57, 15-04, 47N40
Published electronically: January 27, 2011
MathSciNet review: 2775801
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Abstract: We show that positivity, consistency, and the variety condition, which are sufficient to solve the truncated moment problem on planar curves of degree 2, are not sufficient for curves of higher degree. Using new, partly algorithmic, conditions based on positive moment matrix extensions, we present a concrete solution to the truncated moment problem on the curve $ y=x^{3}$. We also use moment matrix extensions to solve (in a less concrete sense) truncated moment problems on curves of the form $ y=g(x)$ and $ yg(x)=1$ ( $ g\in \mathbb{R}[x]$), leading to degree-bounded weighted sum-of-squares representations for polynomials which are strictly positive on such curves.


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Additional Information

Lawrence A. Fialkow
Affiliation: Department of Computer Science, State University of New York, New Paltz, New York 12561
Email: fialkowl@newpaltz.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05262-1
PII: S 0002-9947(2011)05262-1
Keywords: Truncated moment problems, moment matrix extensions, representing measures, positive polynomials
Received by editor(s): April 19, 2008
Received by editor(s) in revised form: July 10, 2009
Published electronically: January 27, 2011
Additional Notes: This research was partially supported by NSF Research Grants DMS-0457138 and DMS-0758378.
Article copyright: © Copyright 2011 American Mathematical Society