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

• 成果类型: Article
• 署名作者: De Jeu, M
• 署名单位: Leiden University; Leiden University - Excl LUMC
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1055425776
• 发表日期: 2003
• 页码: 1205-1227
• 关键词: moment problems

摘要

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 less than or equal to p < infinity. 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 R-n 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.