Given a domain $\Omega$ in $\mathbb{ C}^m$, and a finite set of points $z_1,z_2,\ldots, z_n\in \Omega$ and $w_1,w_2,\ldots, w_n\in \mathbb{ D}$ (the open 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 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 \textit{ 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 Schur class is a naturally associated Banach algebra of functions with a domain. The success of the theory lies in characterizing the Schur class interpolating functions for three domains - the bidisc, the symmetrized bidisc and the annulus - which are affiliated to given kernels.
Categories: math.FA