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

de Jeu, Marcel

Ann. Probab., Tome 31 (2003) no. 1, p. 1205-1227 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2003-07-14
Classification:  Determinate multidimensional measures,  Carleman criterion,  $L_p$-spaces,  multidimensional approximation,  polynomials,  trigonometric functions,  multidimensional quasi-analytic classes,  quasi-analytic weights.,  44A60,  41A63,  41A10,  42A10,  46E30,  26E10