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July 2003 Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights
Marcel de Jeu
Ann. Probab. 31(3): 1205-1227 (July 2003). DOI: 10.1214/aop/1055425776


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.


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Marcel de Jeu. "Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights." Ann. Probab. 31 (3) 1205 - 1227, July 2003.


Published: July 2003
First available in Project Euclid: 12 June 2003

zbMATH: 1050.44003
MathSciNet: MR1988469
Digital Object Identifier: 10.1214/aop/1055425776

Primary: 44A60
Secondary: 26E10 , 41A10 , 41A63 , 42A10 , 46E30

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

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 3 • July 2003
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