矩泛函的变换
Transformations of Moment Functionals
摘要：
在测度论中，有几个结果是已知的，测度空间是如何相互转化的。但是，由于时刻泛函表示的措施，我们在这项研究中调查的影响和影响，这些措施的时刻泛函转换。我们得到了矩泛函的特征。除其他事项外，我们表明，对于一个紧凑的路径连接集$K\subset\mathbb{R}^n$存在一个可测的功能$g:K\to [0,1]$，使任何线性功能$L:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}$是$K$ -时刻功能的当且仅当它有一个连续的扩展到一些$\overline{L}:\mathbb{R}[x_1,\dots,x_n]+\mathbb{R}[g]\to\mathbb{R}$，使$\tilde{L}:\mathbb{R}[t]\to\mathbb{R}$定义的$\tilde{L}(t^d) := \overline{L}(g^d)$为所有$d\in\mathbb{N}_0$是一个$[0,1]$ -时刻功能（Hausdorff时刻问题）。另外，存在独立于$L$的连续函数$f:[0,1]\to K$，使得$\tilde{L}$的表示度量$\tildeμ$提供$L$的表示度量$\tildeμ\circ f^{-1}$。我们还证明了对于某些可测函数$f:[0,1]\to\mathbb{R}^n$，每个矩泛函$L:\mathcal{V}\to\mathbb{R}$都由$λ\circ f^{-1}$表示，其中$λ$是$[0,1]$上的Lebesgue.
英文摘要：
In measure theory several results are known how measure spaces are transformed into each other. But since moment functionals are represented by a measure we investigate in this study the effects and implications of these measure transformations to moment funcationals. We gain characterizations of moments functionals. Among other things we show that for a compact and path connected set $K\subset\mathbb{R}^n$ there exists a measurable function $g:K\to [0,1]$ such that any linear functional $L:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}$ is a $K$-moment functional if and only if it has a continuous extension to some $\overline{L}:\mathbb{R}[x_1,\dots,x_n]+\mathbb{R}[g]\to\mathbb{R}$ such that $\tilde{L}:\mathbb{R}[t]\to\mathbb{R}$ defined by $\tilde{L}(t^d) := \overline{L}(g^d)$ for all $d\in\mathbb{N}_0$ is a $[0,1]$-moment functional (Hausdorff moment problem). Additionally, there exists a continuous function $f:[0,1]\to K$ independent on $L$ such that the representing measure $\tildeμ$ of $\tilde{L}$ provides the representing measure $\tildeμ\circ f^{-1}$ of $L$. We also show that every moment functional $L:\mathcal{V}\to\mathbb{R}$ is represented by $λ\circ f^{-1}$ for some measurable function $f:[0,1]\to\mathbb{R}^n$ where $λ$ is the Lebesgue on $[0,1]$.
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