The Annals of Probability

Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights

Marcel de Jeu

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Abstract

We prove in a direct fashion that a multidimensional probability measure $\mu$ is determinate if the higher-dimensional analogue of Carleman's condition is satisfied. In that case, the polynomials, as well as certain proper subspaces of the trigonometric functions, are dense in all associated $L_p$-spaces for $1\leq p<\infty$. In particular these three statements hold if the reciprocal of a quasi-analytic weight has finite integral under $\mu$. We give practical examples of such weights, based on their classification.

As in the one-dimensional case, the results on determinacy of measures supported on $\Rn$ lead to sufficient conditions for determinacy of measures supported in a positive convex cone, that is, the higher-dimensional analogue of determinacy in the sense of Stieltjes.

Article information

Source
Ann. Probab., Volume 31, Number 3 (2003), 1205-1227.

Dates
First available in Project Euclid: 12 June 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1055425776

Digital Object Identifier
doi:10.1214/aop/1055425776

Mathematical Reviews number (MathSciNet)
MR1988469

Zentralblatt MATH identifier
1050.44003

Subjects
Primary: 44A60: Moment problems
Secondary: 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section) 41A10: Approximation by polynomials {For approximation by trigonometric polynomials, see 42A10} 42A10: Trigonometric approximation 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 26E10: $C^\infty$-functions, quasi-analytic functions [See also 58C25]

Keywords
Determinate multidimensional measures Carleman criterion $L_p$-spaces multidimensional approximation polynomials trigonometric functions multidimensional quasi-analytic classes quasi-analytic weights.

Citation

de Jeu, Marcel. Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights. Ann. Probab. 31 (2003), no. 3, 1205--1227. doi:10.1214/aop/1055425776. https://projecteuclid.org/euclid.aop/1055425776


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