Nonlinear solutions of Nevanlinna-Pick interpolation problems
Let us recall the Nevanlinna-Pick interpolation problem (NP): Given points \(\{z_ 1,...,z_ k\}\) in the open unit disk D in \({\mathbb{C}}\), and two sets of vectors \(\{x_ 1,...,x_ k\}\in {\mathbb{C}}^ n\) and \(\{y_ 1,...,y_ k\}\in {\mathbb{C}}^ m,\) find a matrix valued holomorphic function \(F: D\to M_{m\times n}\) such that \[ i)\quad \| F(z)\| \leq 1\quad for\quad z\in D,\quad and\quad ii)\quad F(z_ j)x_ j=y_ j\quad (1\leq j\leq k). \] In the dual (NP*) problem the vectors are given as rows instead of columns. The main point of this article is that (NP) and (NP*) can be posed for nonlinear functions (i.e., F is not a matrix valued function) and that (NP)\(\cap\)(NP*) has a solution if and only if there is a nonlinear solution. This generalizes work of \textit{P. P. Khargonekar} and \textit{K. R. Poolla} [Syst. Control Lett. 6, 303-308 (1985; Zbl 0594.93032)].
scientific article; zbMATH DE number 4059795