A new algorithm for meromorphic Nevanlinna-Pick interpolation
Let \(H_0^\infty\) be the space of all bounded analytic functions in the unit disk \(D\). The set of all functions \(f(z)/p(z)\), where \(p(z)=(z-z_1)...(z-z_k)\), \(z_i\in D\) and \(f\in H_0^\infty\), is denoted by \(H_k^\infty\). Problem I: Let \(z_j\in D\), \(w_j\in {\mathbb{C}}\), \(j=1,...,n\), and \(0\leq k <n\). Find \(f\in H_k^\infty\) such that \(f(z_i)=w_i\), \(1\leq i\leq n\) and \(\| f\| _\infty\) is minimized. Assumption A: Given the interpolation data Problem I has a solution in each class \(H^\infty_k\), \(0<k<n\). The main result of this paper (Theorem 2.3) gives a detailed description of the rational function which is the unique solution of Problem I under Assumption A. The authors present an algorithm which is derived from Theorem 2.3 for the numerical solution of Problem I. Two examples have been chosen to illustrate the power of this approach.
scientific article; zbMATH DE number 2156013