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Type of Publication: | Journal article |
Author: | Ghasemi, Mehdi; Kuhlmann, Salma |
Year of publication: | 2013 |
Published in: | Journal of Functional Analysis ; 264 (2013), 1. - pp. 413-427. - ISSN 0022-1236. - eISSN 1096-0783 |
DOI (citable link): | https://dx.doi.org/10.1016/j.jfa.2012.10.018 |
Summary: |
Let R be a unitary commutative R-algebra and K⊆X(R)=Hom(R,R), closed with respect to the product topology. We consider R endowed with the topology T(K), induced by the family of seminorms ρα(a):=|α(a)|, for α∈K and a∈R. In case K is compact, we also consider the topology induced by ‖a‖K:=supα∈K|α(a)| for a∈R. If K is Zariski dense, then those topologies are Hausdorff. In this paper we prove that the closure of the cone of sums of 2d-powers, ∑R2d, with respect to those two topologies is equal to Psd(K):={a∈R:α(a)⩾0, for all α∈K}. In particular, any continuous linear functional L on the polynomial ring View the MathML source with L(h2d)⩾0 for each View the MathML source is integration with respect to a positive Borel measure supported on K. Finally we give necessary and sufficient conditions to ensure the continuity of a linear functional with respect to those two topologies.
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Subject (DDC): | 510 Mathematics |
Keywords: | Positive polynomials, Sums of squares, Cone of sums of 2d-powers, Semialgebraic sets, Locally convex topologies, Positive semidefinite continuous linear functionals, Moment problem |
Bibliography of Konstanz: | Yes |
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GHASEMI, Mehdi, Salma KUHLMANN, 2013. Closure of the cone of sums of 2d-powers in real topological algebras. In: Journal of Functional Analysis. 264(1), pp. 413-427. ISSN 0022-1236. eISSN 1096-0783. Available under: doi: 10.1016/j.jfa.2012.10.018
@article{Ghasemi2013Closu-21247, title={Closure of the cone of sums of 2d-powers in real topological algebras}, year={2013}, doi={10.1016/j.jfa.2012.10.018}, number={1}, volume={264}, issn={0022-1236}, journal={Journal of Functional Analysis}, pages={413--427}, author={Ghasemi, Mehdi and Kuhlmann, Salma} }
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