The Nevanlinna-Pick problem on the closed unit disk: minimal norm rational solutions of low degree
A family of rational solutions of degree \(\leq n-1\) for the Nevanlinna-Pick interpolation problem with \(n\) interpolation conditions is constructed. Necessary and sufficient conditions for existence and uniqueness of solutions of this problem with minimal \(H^\infty\)-norm are established and a family of rational solutions of degree \(\leq n-1\) with a minimal norm is constructed in the indeterminate case. It is shown that if the interpolation nodes and the target values are unimodular, then any rational solution of degree \(\leq n-1\) is necessarily a Blaschke product. This expands a previous result by \textit{W. B. Jones} and \textit{S. Ruscheweyh} [Constructive Approximation 3, 405--409 (1987; Zbl 0634.41007)].
scientific article; zbMATH DE number 6035222