Arkiv för Matematik

  • Ark. Mat.
  • Volume 56, Number 2 (2018), 441-459.

On the infinite-dimensional moment problem

Konrad Schmüdgen

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This paper deals with the moment problem on a (not necessarily finitely generated) commutative unital real algebra $A$. We define moment functionals on $A$ as linear functionals which can be written as integrals over characters of $A$ with respect to cylinder measures. Our main results provide such integral representations for $A_{+}$–positive linear functionals (generalized Haviland theorem) and for positive functionals fulfilling Carleman conditions. As an application, we solve the moment problem for the symmetric algebra $S(V)$ of a real vector space $V$. As a byproduct, we obtain new approaches to the moment problem on $S(V)$ for a nuclear space $V$ and to the integral decomposition of continuous positive functionals on a barrelled nuclear topological algebra $A$.

Article information

Ark. Mat., Volume 56, Number 2 (2018), 441-459.

Received: 10 December 2017
Revised: 4 April 2018
First available in Project Euclid: 19 June 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 44A60: Moment problems
Secondary: 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] 46G12: Measures and integration on abstract linear spaces [See also 28C20, 46T12]

moment problem cylinder measure symmetric algebra nuclear space Carleman condition


Schmüdgen, Konrad. On the infinite-dimensional moment problem. Ark. Mat. 56 (2018), no. 2, 441--459. doi:10.4310/ARKIV.2018.v56.n2.a14.

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