AI 中文总结
本文解决实轴与半实轴上的矩阵值单变量有理截断矩问题，给出正矩阵值表示测度的充要条件，通过构造性证明得到相关极小表示测度，其成果可应用于矩阵值矩问题领域。
AI 中文摘要
我们求解实轴和半实轴上的矩阵值单变量有理截断矩问题，给出了存在正矩阵值表示测度的显式充要条件，并证明每个可解问题都存在有限原子表示测度，其总原子重数等于关联矩矩阵的秩。该有理问题被归化为普通矩阵值矩问题，附加要求是表示测度避开有理数据的实极点。主要新要素是一个同时指定节点的结果：有限多个指定点处的最小可达重数可由单个极小表示测度实现。我们的证明是构造性的，使用了平坦延拓、块列关系和定位条件。
英文摘要
We solve the matricial univariate rational truncated moment problem on the real line and on a half-line. We give explicit necessary and sufficient conditions for the existence of a positive matrix-valued representing measure and show that every solvable problem admits a finitely atomic representing measure whose total atomic multiplicity equals the rank of the associated moment matrix. The rational problem is reduced to an ordinary matricial moment problem with the additional requirement that the representing measure avoid the real poles of the rational data. The main new ingredient is a simultaneous prescribed-node result: the smallest attainable multiplicities at finitely many prescribed points can be realized by a single minimal representing measure. Our proofs are constructive and use flat extensions, block-column relations, and localizing conditions.
Comments34 pages
