On the Nevanlinna problem: characterization of all Schur–Agler class solutions affiliated with a given kernel
Volume 255 / 2020
Abstract
Given a domain $\Omega $ in $\mathbb {C}^m$, and finite sets of points $z_1,\ldots , z_n\in \Omega $ and $w_1,\ldots , w_n\in \mathbb {D}$ (the open unit disc in the complex plane), the Pick interpolation problem asks when there is a holomorphic function $f:\Omega \rightarrow \overline {\mathbb {D}}$ such that $f(z_i)=w_i,1\leq i\leq n$. Pick gave a condition on the data $\{z_i, w_i:1\leq i\leq n\}$ 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 a domain $\Omega $ in $\mathbb {C}^m$ admitting holomorphic test functions when the function $f$ comes from the Schur–Agler class and is affiliated with a certain completely positive kernel. The success of the theory lies in characterizing the Schur–Agler class interpolating functions for three domains—the bidisc, the symmetrized bidisc and the annulus—which are affiliated to given kernels.
