Facial behaviour of analytic functions on the bidisc

@article{Agler2011FacialBO,
  title={Facial behaviour of analytic functions on the bidisc},
  author={Jim Agler and John E. McCarthy and N. J. Young},
  journal={Bulletin of the London Mathematical Society},
  year={2011},
  volume={43}
}
We prove that if φ is an analytic function bounded by 1 on the bidisc 픻2 and τ is a point in a face of 픻2 at which φ satisfies Carathéodory's condition, then both φ and the angular gradient ∇φ exist and are constant on the face. Moreover, the class of all φ with prescribed φ(τ) and ∇φ(τ) 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 the faces of the bidisc. 
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