The scalar Nevanlinna-Pick interpolation problem with boundary conditions
The scalar Nevanlinna-Pick interpolation problem with boundary conditions
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具有边界条件的标量 Nevanlinna-Pick 插值问题
DOI：
10.1016/j.cam.2010.11.013
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
P. R. Brown
中科院分区：
文献类型：
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作者：
L. Luxemburg;P. R. Brown
我们证明了如果Nevanlinna-Pick插值问题可由映射到单位圆的紧子集的函数解，则只要边界插值值的模小于1，则在任意数目的边界插值条件下，该问题仍然是可解的.本文给出了在插值区域（右半平面或单位圆盘）的内部和边界上具有任意有限个插值点的Nevanlinna-Pick插值问题的新的归纳证明，并给出了函数值和任意有限个导数的证明.我们的解是关于插值域的闭包的解析解。我们的证明只需要最少的矩阵理论和算子理论。我们还给出了求极小H ∞范数解的新的、直接的算法。 最后给出了数值例子.
We show that if the Nevanlinna–Pick interpolation problem is solvable by a function mapping into a compact subset of the unit disc, then the problem remains solvable with the addition of any number of boundary interpolation conditions, provided the boundary interpolation values have modulus less than unity. We give new, inductive proofs of the Nevanlinna–Pick interpolation problem with any finite number of interpolation points in the interior and on the boundary of the domain of interpolation (the right half plane or unit disc), with function values and any finite number of derivatives specified. Our solutions are analytic on the closure of the domain of interpolation. Our proofs only require a minimum of matrix theory and operator theory. We also give new, straightforward algorithms for obtaining minimal H∞norm solutions. Finally, some numerical examples are given.