Facial behaviour of analytic functions on the bidisk

Details Facial behaviour of analytic functions on the bidisk Facial behaviour of analytic functions on the bidisk

2010-03-17

arxiv.org/abs/1003.3400v2

Mathematics

Complex Variables

18 pages. We have replaced an erroneous proof of Theorem 5.4(1) by a valid proof

Scientific paper

10.1112/blms/bdq115

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.

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