Firstly, we recall the classical moment problem and some basic results related to it. By its formulation, this is an inverse problem: being given a sequence ${\left(y_j\right)}_{j\in {\mathbb{N}}^n}$ of real numbers and a closed subset $F\subseteq {ℝ}^n$, $n\in \left\{1,2,\dots \right\},$ find a positive regular Borel measure $\mu$ on $F$ such that $\underset{F}{\overset{}{{\displaystyle \int }}}{t}^{j}d\mu =y_j, j\in {\mathbb{N}}^n.$ This is the full moment problem. The existence, uniqueness, and construction of the unknown solution $\mu$ are the focus of attention. The numbers $y_j, j\in {\mathbb{N}}^n$ are called the moments of the measure $\mu .$ When a sandwich condition on the solution is required, we have a Markov moment problem. Secondly, we study the existence and uniqueness of the solutions to some full Markov moment problems. If the moments $y_j$are self-adjoint operators, we have an operator-valued moment problem. Related results are the subject of attention. The truncated moment problem is also discussed, constituting the third aim of this work.

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关于当下问题及相关问题

首先，我们回顾一下经典矩问题以及与之相关的一些基本结果。根据它的公式，这是一个逆问题：给定一个序列${\left({是}_j\right)}_{j\in {\mathbb{N}}^n}$ 实数和闭子集 $F\subseteq {ℝ}^n$, $n\in \left\{1,2,\dots \right\},$ 找到正的正则 Borel 测度 $\mu$ 在 $F$ 以至于 $\underset{F}{\overset{}{{\displaystyle \int }}}{吨}^{j}d\mu ={是}_j, j\in {\mathbb{N}}^n.$这是完整的时刻问题。未知解的存在性、唯一性和构造$\mu$是关注的焦点。数字${是}_j, j\in {\mathbb{N}}^n$ 称为度量的矩 $\mu .$当需要解决方案的三明治条件时，我们有一个马尔可夫矩问题。其次，我们研究了一些完全马尔可夫矩问题的解的存在性和唯一性。如果瞬间${是}_j$是自伴随算子，我们有一个算子值矩问题。相关结果是关注的主题。还讨论了截断矩问题，构成了这项工作的第三个目标。