Determinate multidimensional measures, the extended Carleman theorem and quasianalytic weights
Abstract
We prove in a direct fashion that a multidimensional probability measure 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 the associated L_p spaces for all finite p. In particular these three statements hold if the reciprocal of a quasianalytic weight has finite integral under the measure. 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 R^n lead to sufficient conditions for determinacy of measures supported in a positive convex cone, i.e. the higher dimensional analogue of determinacy in the sense of Stieltjes.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 2001
 arXiv:
 arXiv:math/0111019
 Bibcode:
 2001math.....11019D
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Functional Analysis;
 Mathematics  Probability;
 44A60 (Primary) 41A63;
 41A10;
 42A10;
 46E30;
 26E10 (Secondary)
 EPrint:
 20 pages, LaTeX 2e, no figures. Second and final version, with minor corrections and an additional section on Stieltjes determinacy in arbitrary dimension. Accepted by The Annals of Probability