Abstract
The tetrablock is the setE={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↑1x(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.
Highlights
In this paper we present an algorithm for the construction of a general rational inner function from D to the tetrablock
The algorithm is based on a known solution of the Nevanlinna-Pick interpolation problem on D
The first question that arises is: what data should replace the αj, the zeros of φ? We have found that an effective choice is the set of royal nodes of the tetra-inner function, which we shall define
Summary
In this paper we present an algorithm for the construction of a general rational inner function from D to the tetrablock. In Theorem 1.7, we give a prescription for the construction of a general rational tetra-inner function of degree n in terms of its royal nodes and royal values. To describe our main results on the construction of a general rational tetra-inner function we need to recall some definitions and results on the Blaschke interpolation problem. (The royal tetra-interpolation problem) Given royal tetra-interpolation data (σ, η, η, ρ) of type (n, k), find all rational E-inner functions x = (x1, x2, x3) of degree n such that x(σj) = (ηj, ηj, ηjηj) for j = 1, ..., n, and Ax1(σj) = ρj for j = 1, ..., k. In [5] there is a construction of the general rational E-inner function x = (x1, x2, x3) of degree n, in terms of different data, namely, the royal nodes of x and the zeros of x1 and x2.
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