确定的多维测度、扩展的Carleman定理与拟解析权
Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights
AI总结：
本文证明了满足高维Carleman条件的多维概率测度是确定的，并给出拟解析权的实际例子，进而将结果推广到正凸锥上测度的Stieltjes型确定性。
摘要：
我们以一种直接的方式证明：如果一个多维概率测度满足Carleman条件的高维类比，则该测度是确定的。在这种情况下，多项式以及三角函数的某些适当子空间在所有有限p对应的L_p空间中都是稠密的。特别地，如果一个拟解析权的倒数在该测度下具有有限积分，则上述三个结论成立。我们基于这些权的分类给出了实际例子。与一维情形一样，关于支撑在R^n上的测度确定性的结果可以导出支撑在正凸锥上的测度确定性的充分条件，即Stieltjes意义下确定性的高维类比。
英文摘要：
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 quasi-analytic 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.
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