The Moment Problem for Continuous Positive Semidefinite Linear Functionals

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2012
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Ghasemi, Mehdi
Samei, Ebrahim
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Archiv der Mathematik. 2012, 100(1), pp. 43-53. ISSN 0003-889X. eISSN 1420-8938. Available under: doi: 10.1007/s00013-012-0460-5
Zusammenfassung

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.

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Fachgebiet (DDC)
510 Mathematik
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Positive polynomials, Sums of squares, Real algebraic geometry, Moment problem, Weighted norm topologies
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ISO 690GHASEMI, Mehdi, Salma KUHLMANN, Ebrahim SAMEI, 2012. The Moment Problem for Continuous Positive Semidefinite Linear Functionals. In: Archiv der Mathematik. 2012, 100(1), pp. 43-53. ISSN 0003-889X. eISSN 1420-8938. Available under: doi: 10.1007/s00013-012-0460-5
BibTex
@article{Ghasemi2012Momen-21246,
  year={2012},
  doi={10.1007/s00013-012-0460-5},
  title={The Moment Problem for Continuous Positive Semidefinite Linear Functionals},
  number={1},
  volume={100},
  issn={0003-889X},
  journal={Archiv der Mathematik},
  pages={43--53},
  author={Ghasemi, Mehdi and Kuhlmann, Salma and Samei, Ebrahim}
}
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