Boundary behavior of analytic functions of two variables via generalized models

Details Boundary behavior of analytic functions of two variables via generalized models Boundary behavior of analytic functions of two variables via generalized models

2012-03-29

arxiv.org/abs/1203.6589v1

Mathematics

Complex Variables

30 pages

Scientific paper

We describe a generalization of the notion of a Hilbert space model of a function in the Schur class of the bidisc. This generalization is well adapted to the investigation of boundary behavior at a mild singularity of the function on the 2-torus. We prove the existence of a generalized model with certain properties corresponding to such a singularity and use this result to solve two function-theoretic problems. The first of these is to characterise the directional derivatives of a function in the Schur class at a singular point on the torus for which the Carath\'eodory condition holds. The second is to obtain a representation theorem for functions in the two-variable Pick class analogous to the refined Nevanlinna representation of functions in the one-variable Pick class.

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