On the Nevanlinna problem - Characterization of all Schur-Agler class solutions

Abstract

Given a domain $\Omega$ in ${\mathbb{C}}^m$, and a finite set of points $z_1,\dots ,z_n\in \Omega$ and $w_1,\dots ,w_n\in \mathbb{D}$ (the open unit disc in the complex plane), the $Pickinterpolationproblem$ asks when there is a holomorphic function $f:\Omega \to \overline{\mathbb{D}}$ such that $f(z_i)=w_i,1\le i\le n$. Pick gave a condition on the data $\{z_i,w_i:1\le i\le n\}$ for such an $interpolant$ to exist if $\Omega =\mathbb{D}$. Nevanlinna characterized all possible functions $f$ that $interpolate$ the data. We generalize Nevanlinna's result to an arbitrary set $\Omega$. In this case, the function $f$ comes from the Schur-Agler class. The abstract result is then applied to three examples - the bidisc, the symmetrized bidisc and the annulus. In these examples, the Schur-Agler class is the same as the Schur class.

The article has some serious mistakes