Interpolation by holomorphic maps from the disc to the tetrablock

Abstract

The tetrablock is the set $$\mathcal{E}=\{x\in {\mathbb{C}}^3:1-x_{1}z-x_{2}w+x_{3}zw\ne 0whenever|z|\le 1,|w|\le 1\}.$$ 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 $\mathbb{T}$, \[ \lim_{r\uparrow 1} x(r λ) \mbox{ exists and lies in } b \overline{\mathcal{E}}, \] 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.

35 pages, the paper has been accepted for publication in the Journal of Mathematical Analysis and Applications