Article ID Journal Published Year Pages File Type
4672877 Indagationes Mathematicae 2012 33 Pages PDF
Abstract

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 characterize the directional derivatives of a function in the Schur class at a singular point on the torus for which the Carathéodory 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.

Related Topics
Physical Sciences and Engineering Mathematics Mathematics (General)
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