Interpolation by holomorphic maps from the disc to the tetrablock

Abstract

The tetrablock is the set E={x∈C3:1−x1z−x2w+x3zw≠0whenever|z|≤1,|w|≤1}. The closure of E is denoted by E‾. A tetra-inner function is an analytic map x from the unit disc D to E‾ such that, for almost all points λ of the unit circle T, limr↑1⁡x(rλ) exists and lies in bE‾, where bE‾ denotes the distinguished boundary of 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|T from T to bE‾. 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=(x1,x2,x3) is a rational tetra-inner function of degree n, then x1x2−x3 either is identically 0 or has precisely n zeros in the closed unit disc D‾, counted with multiplicity. It turns out that a natural choice of data for the construction of a rational tetra-inner function x=(x1,x2,x3) consists of the points in D‾ for which x1x2−x3=0 and the values of x at these points.