Archiv der Mathematik

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

The moment problem for continuous positive semidefinite linear functionals



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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  1. 1.
    Berg C., Christensen J.P.R., Jensen C.U.: A remark on the multidimensional moment problem, Math. Ann. 243, 163–169 (1979)MathSciNetMATHGoogle Scholar
  2. 2.
    Berg C., Christensen J.P.R., Ressel P.: Positive definite functions on abelian semigroups, Math. Ann. 223, 253–274 (1976)MathSciNetMATHGoogle Scholar
  3. 3.
    C. Berg, J. P. R. Christensen, and P. Ressel, Harmonic Analysis on Semigroups, Theory of Positive Definite and Related Functions, Springer-Verlag, (1984).Google Scholar
  4. 4.
    Berg C., Maserick P.H.: Exponentially bounded positive definite functions, Illinois J. Math. 28, 162–179 (1984)MathSciNetMATHGoogle Scholar
  5. 5.
    J. B. Conway, A Course in Functional Analysis, 2nd ed., GTM. 96, Springer (1990).Google Scholar
  6. 6.
    M. Ghasemi and S. Kuhlmann, Closure of the cone of sums of 2d-powers in real topological algebras, to appear in J. Funct. Anal.Google Scholar
  7. 7.
    M. Ghasemi, M. Marshall, and S. Wagner, Closure of the cone of sums of 2d-powers in certain weighted ℓ1-seminorm topologies, to appear.Google Scholar
  8. 8.
    Haviland E.K.: On the momentum problem for distribution functions in more than one dimension, Amer. J. Math. 57, 562–572 (1935)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Haviland E.K.: On the momentum problem for distribution functions in more than one dimension II, Amer. J. Math. 58, 164–168 (1936)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Krivine J.L.: Anneaux préordonnés, J. Analyse Math. 167, 160–196 (1932)Google Scholar
  11. 11.
    Kuhlmann S., Marshall M.: Positivity, sums of squares and the multi-dimensional moment problem, Trans. Amer. Math. Soc. 354, 4285–4301 (2002)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Lasserre J.B.: Global optimization with polynomials and the problem of moments, SIAM J. Optimization 11, 796–817 (2001)MathSciNetMATHGoogle Scholar
  13. 13.
    J. B. Lasserre, The K-Moment problem for continuous linear functionals, To appear in Trans. Amer. Math. (2011).Google Scholar
  14. 14.
    Lasserre J.B., Netzer T.: SOS approximation of nonnegative polynomials via simple high degree perturbations, Math. Zeitschrift 256, 99–112 (2006)MathSciNetGoogle Scholar
  15. 15.
    M. Marshall, Positive Polynomials and Sum of Squares, Mathematical Surveys and Monographs, Vol 146. (2007).Google Scholar
  16. 16.
    Putinar M.: Positive polynomials on compact semialgebraic sets, Indiana Univ. Math. J. 43, 969–984 (1993)MathSciNetGoogle Scholar
  17. 17.
    W. Rudin, Functional Analysis, 2nd ed., International series in pure and applied mathematics, (1991).Google Scholar

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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