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

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.