广义逆本征值问题与$m$ -函数
A generalized inverse eigenvalue problem and $m$-functions
摘要：
本文考虑了一类广义逆本征值问题，它涉及有理插值和双正交有理函数理论中的线性矩阵束$(z\mathcal{J}_{[0,n]}-\mathcal{H}_{[0,n]})$.除了重建的埃尔米特矩阵$\mathcal{H}_{[0,n]}$的条目$b_j's$，有理函数的特征，是指定的特征向量的分量。关于$\mathcal{J}_{[0,n]}$的正定性的条件，这往往是一个假设，在直接的问题也是孤立的。此外，$\mathcal{H}_{[0,n]}$的重建是通过铅笔$(z\mathcal{J}_{[0,n]}-\mathcal{H}_{[0,n]})$的逆来观察的，其涉及$m$函数的概念。
英文摘要：
In this manuscript, a generalized inverse eigenvalue problem is considered that involves a linear pencil $(z\mathcal{J}_{[0,n]}-\mathcal{H}_{[0,n]})$ of matrices arising in the theory of rational interpolation and biorthogonal rational functions. In addition to the reconstruction of the Hermitian matrix $\mathcal{H}_{[0,n]}$ with the entries $b_j's$, characterizations of the rational functions that are components of the prescribed eigenvectors are given. A condition concerning the positive-definiteness of $\mathcal{J}_{[0,n]}$ and which is often an assumption in the direct problem is also isolated. Further, the reconstruction of $\mathcal{H}_{[0,n]}$ is viewed through the inverse of the pencil $(z\mathcal{J}_{[0,n]}-\mathcal{H}_{[0,n]})$ which involves the concept of $m$-functions.
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