Finite Blaschke products and the construction of rational $\Gamma$-inner functions

TL;DR

This paper constructs rational b5-inner functions mapping the unit disk to a specific complex domain, using finite Blaschke products, and characterizes their zeros and boundary behavior.

Contribution

It provides an explicit construction method for rational b5-inner functions with prescribed zeros and boundary values, expanding understanding of their structure.

Findings

Explicit construction of degree n rational b5-inner functions

Characterization of zeros of s^2-4p in relation to the function degree

Use of finite Blaschke products to solve interpolation problems

Abstract

Let \[ \Gamma = \{(z+w, zw): |z|\leq 1, |w|\leq 1\} \subset \mathbb{C}^2. \] A $\Gamma$-inner function is defined to be a holomorphic map $h$ from the unit disc $\mathbb{D}$ to $\Gamma$ whose boundary values at almost all points of the unit circle $\mathbb{T}$ belong to the distinguished boundary $b\Gamma$ of $\Gamma$. A rational $\Gamma$-inner function $h$ induces a continuous map $h|_\mathbb{T}$ from the unit circle to $b\Gamma$. The latter set is topologically a M\"obius band and so has fundamental group $\mathbb{Z}$. The {\em degree} of $h$ is defined to be the topological degree of $h|_\mathbb{T}$. In a previous paper the authors showed that if $h=(s,p)$ is a rational $\Gamma$-inner function of degree $n$ then $s^2-4p$ has exactly $n$ zeros in the closed unit disc $\mathbb{D}^-$, counted with an appropriate notion of multiplicity. In this paper, with the aid of a solution of an interpolation problem for finite Blaschke products, we explicitly construct the rational $\Gamma$-inner functions of degree $n$ with the $n$ zeros of $s^2-4p$ and the corresponding values of $s$, prescribed.