Extremal holomorphic maps and the symmetrised bidisc

TL;DR

This paper introduces a new class of extremal holomorphic maps, explores their application to interpolation problems in the symmetrised bidisc, and proposes a conjecture on necessary and sufficient conditions for solvability.

Contribution

It defines the class of n-extremal holomorphic maps, introduces a sequence of necessary conditions for interpolation solvability, and classifies rational Gamma-inner functions.

Findings

Necessary condition C_{n-3} is insufficient for n-point problems with n≥3.

The sequence C_ν is strictly increasing in strength.

Conjecture: C_{n-2} is necessary and sufficient for n-point interpolation.

Abstract

We introduce the class of $n$-extremal holomorphic maps, a class that generalises both finite Blaschke products and complex geodesics, and apply the notion to the finite interpolation problem for analytic functions from the open unit disc into the symmetrised bidisc $\Gamma$. We show that a well-known necessary condition for the solvability of such an interpolation problem is not sufficient whenever the number of interpolation nodes is 3 or greater. We introduce a sequence $\mathcal{C}_\nu, \nu \geq 0,$ of necessary conditions for solvability, prove that they are of strictly increasing strength and show that $\mathcal{C}_{n-3}$ is insufficient for the solvability of an $n$-point problem for $n\geq 3$. We propose the conjecture that condition $\mathcal{C}_{n-2}$ is necessary and sufficient for the solvability of an $n$-point interpolation problem for $\Gamma$ and we explore the implications of this conjecture. We introduce a classification of rational $\Gamma$-inner functions, that is, analytic functions from the disc into $\Gamma$ whose radial limits at almost all points on the unit circle lie in the distinguished boundary of $\Gamma$. The classes are related to $n$-extremality and the conditions $\mathcal{C}_\nu$; we prove numerous strict inclusions between the classes.