广义函数半代数集上的全infinite dimensional矩问题
The full infinite dimensional moment problem on semi-algebraic sets of generalized functions
摘要：
我们考虑一个通用的基本半代数子集$\mathcal{S}$的广义函数空间，这是一个集（不一定可数多）多项式约束。给出了广义函数的无穷序列在$\mathcal{S}$上可实现的充要条件，即是集中在$\mathcal{S}$上的有限测度的矩序列.我们的方法结合了核型空间上矩问题的经典结果和最近发展起来的处理$\mathbb{R}^d$的基本半代数集上的矩问题的技术。通过这种方式，我们确定的可实现性条件，可以更容易地验证比著名的哈维兰型条件。我们的结果完全刻画了实现测度的支集的矩。作为广义函数的半代数集的具体例子，我们考虑了所有Radon测度的集合和所有Radon-Nikodym密度有界的测度的集合。勒贝格测度。
英文摘要：
We consider a generic basic semi-algebraic subset $\mathcal{S}$ of the space of generalized functions, that is a set given by (not necessarily countably many) polynomial constraints. We derive necessary and sufficient conditions for an infinite sequence of generalized functions to be realizable on $\mathcal{S}$, namely to be the moment sequence of a finite measure concentrated on $\mathcal{S}$. Our approach combines the classical results about the moment problem on nuclear spaces with the techniques recently developed to treat the moment problem on basic semi-algebraic sets of $\mathbb{R}^d$. In this way, we determine realizability conditions that can be more easily verified than the well-known Haviland type conditions. Our result completely characterizes the support of the realizing measure in terms of its moments. As concrete examples of semi-algebraic sets of generalized functions, we consider the set of all Radon measures and the set of all the measures having bounded Radon-Nikodym density w.r.t. the Lebesgue measure.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
