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

 

The $ \mathbf{K}$-moment problem for continuous linear functionals


Author: Jean B. Lasserre
Journal: Trans. Amer. Math. Soc. 365 (2013), 2489-2504
MSC (2010): Primary 44A60, 13B25, 14P10, 30C10
Published electronically: October 4, 2012
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Abstract: Given a closed (and not necessarily compact) basic semi-algebraic set $ \mathbf {K}\subseteq \mathbb{R}^n$, we solve the $ \mathbf {K}$-moment problem for continuous linear functionals. Namely, we introduce a weighted $ \ell _1$-norm $ \ell _{\mathbf {w}}$ on $ \mathbb{R}[\mathbf {x}]$, and show that the $ \ell _{\mathbf {w}}$-closures of the preordering $ P$ and quadratic module $ Q$ (associated with the generators of $ \mathbf {K}$) is the cone $ \textrm {Psd}(\mathbf {K})$ of polynomials nonnegative on $ \mathbf {K}$. We also prove that $ P$ and $ Q$ solve the $ \mathbf {K}$-moment problem for $ \ell _{\mathbf {w}}$-continuous linear functionals and completely characterize those $ \ell _{\mathbf {w}}$-continuous linear functionals nonnegative on $ P$ and $ Q$ (hence on $ \textrm {Psd}(\mathbf {K})$). When $ \mathbf {K}$ has a nonempty interior, we also provide in explicit form a canonical $ \ell _{\mathbf {w}}$-projection $ g^{\mathbf {w}}_f$ for any polynomial $ f$, on the (degree-truncated) preordering or quadratic module. Remarkably, the support of $ g^{\mathbf {w}}_f$ is very sparse and does not depend on $ \mathbf {K}$! This enables us to provide an explicit Positivstellensatz on $ \mathbf {K}$. And last but not least, we provide a simple characterization of polynomials nonnegative on $ \mathbf {K}$, which is crucial in proving the above results.


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Additional Information

Jean B. Lasserre
Affiliation: LAAS-CNRS and Institute of Mathematics, University of Toulouse, LAAS, 7 avenue du Colonel Roche, 31077 Toulouse Cédex 4, France
Email: lasserre@laas.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05701-1
PII: S 0002-9947(2012)05701-1
Keywords: Moment problems, real algebraic geometry, positive polynomials, semi-algebraic sets
Received by editor(s): July 19, 2011
Received by editor(s) in revised form: September 5, 2011, and September 7, 2011
Published electronically: October 4, 2012
Article copyright: © Copyright 2012 American Mathematical Society