Abstract
The tetrablock is the set
epsilon = {x is an element of C-3 : 1- x(1)z -x(2)w + x(3)zw not equal 0 whenever vertical bar z vertical bar <= 1, vertical bar w vertical bar <= 1}.
The closure of epsilon is denoted by (epsilon) over bar. A tetra-inner function is an analytic map x from the unit disc D to (epsilon) over bar such that, for almost all points lambda of the unit circle T,
lim(r up arrow 1) x(r lambda) exists and lies in b (epsilon) over bar,
where b (epsilon) over bar denotes the distinguished boundary of (epsilon) over bar. 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 vertical bar T from T to b (epsilon) over bar. 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 (D) over bar, 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 (D) over bar for which x(1) x(2) - x(3) = 0 and the values of x at these points. (C) 2021 The Author(s). Published by Elsevier Inc.