Archiv der Mathematik

, Volume 100, Issue 1, pp 43–53 | Cite as

The moment problem for continuous positive semidefinite linear functionals

  • Mehdi Ghasemi
  • Salma Kuhlmann
  • Ebrahim Samei


Let τ be a locally convex topology on the countable dimensional polynomial \({\mathbb{R}}\) -algebra \({\mathbb{R} [\underline{X}] := \mathbb{R} [X_1, \ldots, X_{n}]}\) . Let K be a closed subset of \({\mathbb{R} ^{n}}\) , and let \({M := M_{\{g_1, \ldots, g_s\}}}\) be a finitely generated quadratic module in \({\mathbb{R} [\underline{X}]}\) . We investigate the following question: When is the cone Psd(K) (of polynomials nonnegative on K) included in the closure of M? We give an interpretation of this inclusion with respect to representing continuous linear functionals by measures. We discuss several examples; we compute the closure of \({M = \sum \mathbb{R} [\underline{X}]^{2}}\) with respect to weighted norm-p topologies. We show that this closure coincides with the cone Psd(K) where K is a certain convex compact polyhedron.

Mathematics Subject Classification (2010)

Primary 14P99 44A60 Secondary 12D15 43A35 46B99 


Positive polynomials Sums of squares Real algebraic geometry Moment problem Weighted norm topologies 


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Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of SaskatchewanSaskatoonCanada
  2. 2.Fachbereich Mathematik und StatistikUniversitãt KonstanzKonstanzGermany

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