Facial behaviour of analytic functions on the bidisk

TL;DR

This paper characterizes the boundary behavior of bounded analytic functions on the bidisk, showing that under certain conditions, these functions and their gradients are constant on faces, and provides a parametrization and interpolation solutions.

Contribution

It establishes boundary regularity results for analytic functions on the bidisk and introduces a parametrization using the two-variable Pick class, solving related interpolation problems.

Findings

Functions and gradients are constant on faces under Caratheodory's condition.

Provides a parametrization of functions with prescribed boundary data.

Solves an interpolation problem with nodes on faces of the bidisk.

Abstract

We prove that if $\phi$ is an analytic function bounded by 1 on the bidisk and $\tau$ is a point in a face of the bidisk at which $\phi$ satisfies Caratheodory's condition then both $\phi$ and the angular gradient $\nabla\phi$ exist and are constant on the face. Moreover, the class of all $\phi$ with prescribed $\phi(\tau)$ and $\nabla\phi(\tau)$ can be parametrized in terms of a function in the two-variable Pick class. As an application we solve an interpolation problem with nodes that lie on faces of the bidisk.