Finite Blaschke products and the construction of rational Γ-inner functions

Abstract

A Γ-inner function is a holomorphic map h from the unit disc D to Γ whose boundary values at almost all points of the unit circle T belong to the distinguished boundary bΓ of Γ. A rational Γ-inner function h induces a continuous map h| T from T to bΓ. The latter set is topologically a Möbius band and so has fundamental group Z. The degree of h is defined to be the topological degree of h| T . In a previous paper the authors showed that if h = (s, p) is a rational Γ-inner function of degree n then s 2 − 4p has exactly n zeros in the closed unit disc D − , counted with an appropriate notion of multiplicity. In this paper, with the aid of a solution of an interpolation problem for finite Blaschke products, we explicitly construct the rational Γ-inner functions of degree n with the n zeros of s 2 − 4p prescribed.

Interpolation by holomorphic maps from the disc to the tetrablock

“…In this section, for the convenience of the reader, we collect some known facts about finite Blaschke products that we need. They may be found in several places, but the most economical source for our purposes is [3], which assembles precisely the results which we require.…”

“…The Blaschke interpolation Problem 1.3 as described in [3] is an algebraic variant of the classical Pick interpolation problem. One looks for a Blaschke product of degree n satisfying n interpolation conditions, rather than a Schur-class function as in the original Pick interpolation problem.…”

“…In [3] the authors described a strategy for the construction of the general solution of the Blaschke interpolation problem (Problem 1.…”

Positive Polynomials and Boundary Interpolation with Finite Blaschke Products

“…is a rational parametrization of the unit circle, more precisely, it is a bijective mapping from R to {(x, y) ∈ R 2 : (1,0). We also use σ = s/(1 − s) and τ = u/(1 − u 2 ), so if s runs over the interval (0, 1), then σ runs over the interval (0, ∞) and if u runs over the interval (−1, 1), then τ runs over the real numbers.…”

“…Actually, there are several proofs of this result and there is a very nice overview in the excellent paper [17] by Semmler and Wegert from 2006. Some earlier references are (not exhaustive list): [1,[6][7][8][9][10][11][12][13]18,19] and it is also worth mentioning the books [4,5].…”