The multidimensional truncated moment problem: Carathéodory numbers

Let A be a finite-dimensional subspace of C(X;R), where X is a locally compact Hausdorff space, and A={f1,…,fm} a basis of A. A sequence s=(sj)j=1m is called a moment sequence if sj=∫fj(x)dμ(x), j=1,…,m, for some positive Radon measure μ on X. Each moment sequence s has a finitely atomic representin...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2018-05, Vol.461 (2), p.1606-1638
Main Authors: di Dio, Philipp J., Schmüdgen, Konrad
Format: Article
Language:English
Subjects:
Carathéodory number
Convex cone
Positive polynomials
Truncated moment problem
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Summary:Let A be a finite-dimensional subspace of C(X;R), where X is a locally compact Hausdorff space, and A={f1,…,fm} a basis of A. A sequence s=(sj)j=1m is called a moment sequence if sj=∫fj(x)dμ(x), j=1,…,m, for some positive Radon measure μ on X. Each moment sequence s has a finitely atomic representing measure μ. The smallest possible number of atoms is called the Carathéodory number CA(s). The largest number CA(s) among all moment sequences s is the Carathéodory number CA. In this paper the Carathéodory numbers CA(s) and CA are studied. In the case of differentiable functions methods from differential geometry are used. The main emphasis is on real polynomials. For a large class of spaces of polynomials in one variable the number CA is determined. In the multivariate case we obtain some lower bounds and we use results on zeros of positive polynomials to derive upper bounds for the Carathéodory numbers.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2017.12.021