Interpolation by holomorphic maps from the disc to the tetrablock

Abstract

The tetrablock is the set $$ \mathcal{E}=\{x \in \mathbb{C}^3: \quad 1-x_1z-x_2w+x_3z w \neq 0 \quad whenever \quad |z|\leq 1, |w|\leq 1\}. $$ The closure of $\mathcal{E}$ is denoted by $\overline{\mathcal{E}}$. A tetra-inner function is an analytic map $x$ from the unit disc $ \mathbb{D} $ to $\overline{\mathcal{E}}$ such that, for almost all points $\lambda$ of the unit circle $ \mathbb{T}$, \[ \lim_{r\uparrow 1} x(r \lambda) \mbox{ exists and lies in } b \overline{\mathcal{E}}, \] where $b \overline{\mathcal{E}}$ denotes the distinguished boundary of $\overline{\mathcal{E}}$. There is a natural notion of degree of a rational tetra-inner function $ x$; it is simply the topological degree of the continuous map $ x|_\mathbb{T} $ from $ \mathbb{T} $ to $ b \overline{\mathcal{E}} $. In this paper we give a prescription for the construction of a general rational tetra-inner function of degree $n$. The prescription exploits a known construction of the finite Blaschke products of given degree which satisfy some interpolation conditions with the aid of a Pick matrix formed from the interpolation data. It is known that if $x= (x_1, x_2, x_3)$ is a rational tetra-inner function of degree $n$, then $x_1 x_2 - x_3$ either is identically $0$ or has precisely $n$ zeros in the closed unit disc $\overline{\mathbb{D}}$, counted with multiplicity. It turns out that a natural choice of data for the construction of a rational tetra-inner function $x= (x_1, x_2, x_3)$ consists of the points in $\overline{\mathbb{D}}$ for which $x_1 x_2 - x_3=0$ and the values of $x$ at these points.