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The full infinite dimensional moment problem on semi-algebraic sets of generalized functions

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2014

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Kuna, Tobias
Rota, Aldo

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https://doi.org/10.1016/j.jfa.2014.06.012
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http://arxiv.org/abs/1311.2175v4

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Journal of Functional Analysis. 2014, 267(5), pp. 1382-1418. ISSN 0022-1236. eISSN 1096-0783. Available under: doi: 10.1016/j.jfa.2014.06.012

Zusammenfassung

We consider a generic basic semi-algebraic subset S of the space of generalized functions, that is a set given by (not necessarily countably many) polynomial constraints. We derive necessary and sufficient conditions for an infinite sequence of generalized functions to be realizable on S, namely to be the moment sequence of a finite measure concentrated on S. Our approach combines the classical results about the moment problem on nuclear spaces with the techniques recently developed to treat the moment problem on basic semi-algebraic sets of Rd. In this way, we determine realizability conditions that can be more easily verified than the well-known Haviland type conditions. Our result completely characterizes the support of the realizing measure in terms of its moments. As concrete examples of semi-algebraic sets of generalized functions, we consider the set of all Radon measures and the set of all the measures having bounded Radon–Nikodym density w.r.t. the Lebesgue measure.

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Fachgebiet (DDC)
510 Mathematik

Schlagwörter

Moment problem, Realizability, Infinite dimensional, moment problem, Semi-algebraic set

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ISO 690INFUSINO, Maria, Tobias KUNA, Aldo ROTA, 2014. The full infinite dimensional moment problem on semi-algebraic sets of generalized functions. In: Journal of Functional Analysis. 2014, 267(5), pp. 1382-1418. ISSN 0022-1236. eISSN 1096-0783. Available under: doi: 10.1016/j.jfa.2014.06.012
BibTex
@article{Infusino2014infin-33316,
  year={2014},
  doi={10.1016/j.jfa.2014.06.012},
  title={The full infinite dimensional moment problem on semi-algebraic sets of generalized functions},
  number={5},
  volume={267},
  issn={0022-1236},
  journal={Journal of Functional Analysis},
  pages={1382--1418},
  author={Infusino, Maria and Kuna, Tobias and Rota, Aldo}
}
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    <dcterms:abstract xml:lang="eng">We consider a generic basic semi-algebraic subset S of the space of generalized functions, that is a set given by (not necessarily countably many) polynomial constraints. We derive necessary and sufficient conditions for an infinite sequence of generalized functions to be realizable on S, namely to be the moment sequence of a finite measure concentrated on S. Our approach combines the classical results about the moment problem on nuclear spaces with the techniques recently developed to treat the moment problem on basic semi-algebraic sets of R&lt;sup&gt;d&lt;/sup&gt;. In this way, we determine realizability conditions that can be more easily verified than the well-known Haviland type conditions. Our result completely characterizes the support of the realizing measure in terms of its moments. As concrete examples of semi-algebraic sets of generalized functions, we consider the set of all Radon measures and the set of all the measures having bounded Radon–Nikodym density w.r.t. the Lebesgue measure.</dcterms:abstract>
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