The Nevanlinna–Pick problem on the closed unit disk: Minimal norm rational solutions of low degree

摘要

For the Nevanlinna–Pick interpolation problem with n interpolation conditions (interior and boundary), we construct a family of rational solutions of degree at most n−1. We also establish necessary and sufficient conditions for the existence and the uniqueness of a solution with the minimally possible H∞-norm and construct a family of minimal-norm rational solutions of degree at most n−1 in the indeterminate case. Finally, we supplement a result of Ruscheweyh and Jones showing that in case the interpolation nodes and the target values are all unimodular, any rational solution of degree at most n−1 is necessarily a finite Blaschke product.