指标集上带缺口的截断Hamburger矩问题
The truncated Hamburger moment problems with gaps in the index set
摘要：
在这篇文章中，我们解决了四个特殊情况下的截断汉堡矩问题（THMP）的程度$2k$与一个或两个失踪的时刻序列。作为推论，通过适当的替换，我们得到了二元截断矩问题的解的次数为$2k$的特殊曲线。也就是说，对于曲线$y=x^3$（首先由Fialkow求解）、$y^2=x^3$、$y=x^4$，其中已知某个阶矩$2k+1$，并且给出具有某个阶矩的$y^3=x^4$。其主要技术是完成部分正半定矩阵（ppsd），使Curto和Fialkow的THMP的解决方案的条件得到满足。主要工具是使用的性质的半正定汉克尔矩阵和一个结果的PPSD矩阵与一个未知的条目，证明了使用舒尔补充$2\times 2$和$3\times 3$块矩阵的所有完成。
英文摘要：
In this article we solve four special cases of the truncated Hamburger moment problem (THMP) of degree $2k$ with one or two missing moments in the sequence. As corollaries we obtain, by using appropriate substitutions, the solutions to bivariate truncated moment problems of degree $2k$ for special curves. Namely, for the curves $y=x^3$ (first solved by Fialkow), $y^2=x^3$, $y=x^4$ where a certain moment of degree $2k+1$ is known and $y^3=x^4$ with a certain moment given. The main technique is the completion of the partial positive semidefinite matrix (ppsd) such that the conditions of Curto and Fialkow's solution of the THMP are satisfied. The main tools are the use of the properties of positive semidefinite Hankel matrices and a result on all completions of a ppsd matrix with one unknown entry, proved by the use of the Schur complements for $2\times 2$ and $3\times 3$ block matrices.
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