Boundary behavior of analytic functions of two variables via generalized models

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.