Solution of the truncated moment problem with variety $y = x^{3}$
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- by Lawrence A. Fialkow PDF
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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.References
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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
- 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.
- © Copyright 2011 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 363 (2011), 3133-3165
- MSC (2000): Primary 47A57, 47A20, 44A60; Secondary 15A57, 15-04, 47N40
- DOI: https://doi.org/10.1090/S0002-9947-2011-05262-1
- MathSciNet review: 2775801