Extremal solutions of Nevalinna-Pick problems and certain classes of inner functions

TL;DR

This paper investigates the extremal solutions of scaled Nevanlinna-Pick interpolation problems, showing that under certain conditions, these solutions belong to specific classes of inner functions, including Hardy space derivatives, exponential, and alpha-Blaschke products.

Contribution

It establishes that extremal solutions of Nevanlinna-Pick problems are contained within particular classes of inner functions when the associated Blaschke product meets certain criteria.

Findings

Extremal solutions lie in the same class as the Blaschke product under specific conditions.

The paper considers three classical classes: Hardy space derivatives, exponential Blaschke products, and alpha-Blaschke products.

Most extremal solutions are shown to belong to these classes when the Blaschke product is in the respective class.

Abstract

Consider a scaled Nevanlinna-Pick interpolation problem and let $\Pi$ be the Blaschke product whose zeros are the nodes of the problem. It is proved that if $\Pi$ belongs to a certain class of inner functions, then the extremal solutions of the problem or most of them, are in the same class. Three different classical classes are considered: inner functions whose derivative is in a certain Hardy space, exponential Blaschke products and also the well known class of $\alpha$-Blaschke products, for $0<\alpha<1$.