论文标题
力矩功能的转换
Transformations of Moment Functionals
论文作者
论文摘要
在测量理论中，几个结果是如何将测量空间转换为彼此的。但是，由于力矩功能由一项度量表示，我们在本研究中研究了这些测量转换对矩弹性的影响和含义。我们获得矩功能的特征。除其他事项外，我们还表明，对于紧凑型和路径，设置$ k \ subset \ mathbb {r}^n $存在可测量的函数$ g：k \ to [0,1] $，这样任何线性函数$ l：\ m athbb {r} r}它具有连续扩展到某些$ \叠加{l}：\ mathbb {r} [x_1，\ dots，x_n]+\ mathbb {r} [g] \ to \ mathbb {r {r {r} $，因此$ \ tilde {l} $ \ tilde {l}（t^d）：= \ edimelline {l}（g^d）$ for \ in \ mathbb {n} _0 $ is a $ [0,1] $ - 力矩函数（hausdorff moment问题）。此外，存在连续函数$ f：[0,1] \ to k $独立于$ l $上的$ \tildeμ$的$ \ tilde {l} $提供表示表示的度量$ \tildeμ\ circ \ circ f^{ - 1} $ $ l $。我们还表明，对于某些可测量函数$ f：[0,1] \ to \ circ f^{ - 1} $表示，每时每刻的功能$ l：\ mathcal {v} \ to \ mathbb {r} $用于某些可测量函数$ f：[0,1] \ to \ mathbb {r Mathbb {r}^n $ where $λ$是$ [0,1] $ [0,1]
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 $[0,1]$.
