带间隙和不带间隙的强截断Hamburger矩问题
The strong truncated Hamburger moment problem with and without gaps
摘要：
强截断汉堡矩问题（STHMP）的程度$(-2k_1,2k_2)$要求找到一个正的Borel测度存在的必要和充分条件，支持$\mathbb{R}\setminus \{0\}$，使$β_i=\int x^idμ\; (-2k_1\leq i\leq 2k_2)$ .利用截断Hamburger矩问题的解和Hankel矩阵的性质，我们求解了STHMP。然后，利用STHMP与$(-2k,2k)$的等价性，得到了Curto和Fialkow首次求解的二维$2k$阶截断矩问题（TMP）的解，该问题具有$xy=1$簇.我们除了他们的结果是以前只知道$k=2$的事实，即存在的措施是等价的存在一个平坦的扩展的时刻矩阵。进一步地，我们解决了序列中缺少一个时刻的次数为$(-2k_1,2k_2)$的STHMP，即，$β_{-2k_1+1}$或$β_{2k_2-1}$，它也给出了TMP的解决方案与品种$x^2y=1$作为一个特殊的情况下，首先由Fialkow研究。
英文摘要：
The strong truncated Hamburger moment problem (STHMP) of degree $(-2k_1,2k_2)$ asks to find necessary and sufficient conditions for the existence of a positive Borel measure, supported on $\mathbb{R}\setminus \{0\}$, such that $β_i=\int x^idμ\; (-2k_1\leq i\leq 2k_2)$. Using the solution of the truncated Hamburger moment problem and the properties of Hankel matrices we solve the STHMP. Then, using the equivalence with the STHMP of degree $(-2k,2k)$, we obtain the solution of the 2-dimensional truncated moment problem (TMP) of degree $2k$ with variety $xy=1$, first solved by Curto and Fialkow. Our addition to their result is the fact previously known only for $k=2$, that the existence of a measure is equivalent to the existence of a flat extension of the moment matrix. Further on, we solve the STHMP of degree $(-2k_1,2k_2)$ with one missing moment in the sequence, i.e., $β_{-2k_1+1}$ or $β_{2k_2-1}$, which also gives the solution of the TMP with variety $x^2y=1$ as a special case, first studied by Fialkow.
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