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Given a domain $\Omega$ in $\mathbb{C}^m$, and a finite set of points$z_1,z_2,\ldots, z_n\in \Omega$ and $w_1,w_2,\ldots, w_n\in \mathbb{D}$ (theopen unit disc in the complex plane), the \textit{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 acondition on the data $\{z_i, w_i:1\leq i\leq n\}$ for such an $interpolant$ toexist if $\Omega=\mathbb{D}$. Nevanlinna characterized all possible functions$f$ that \textit{interpolate} the data. We generalize Nevanlinna's result to adomain $\Omega$ in $\mathbb{C}^m$ admitting holomorphic test functions when thefunction $f$ comes from the Schur-Agler class and is affiliated with a certaincompletely positive kernel. The Schur class is a naturally associated Banachalgebra of functions with a domain. The success of the theory lies incharacterizing the Schur class interpolating functions for three domains - thebidisc, the symmetrized bidisc and the annulus - which are affiliated to givenkernels.

On the Nevanlinna problem -- Characterization of all Schur-Agler class solutions affiliated with a given kernel