Nevanlinna–Pick interpolation

In complex analysis, given initial data consisting of ${\displaystyle n}$ points ${\displaystyle \lambda _{1},\ldots ,\lambda _{n}}$ in the complex unit disc ${\displaystyle \mathbb {D} }$ and target data consisting of ${\displaystyle n}$ points ${\displaystyle z_{1},\ldots ,z_{n}}$ in ${\displaystyle \mathbb {D} }$, the Nevanlinna–Pick interpolation problem is to find a holomorphic function ${\displaystyle \varphi }$ that interpolates the data, that is for all ${\displaystyle i\in \{1,...,n\}}$,

${\displaystyle \varphi (\lambda _{i})=z_{i}}$,

subject to the constraint ${\displaystyle \left\vert \varphi (\lambda )\right\vert \leq 1}$ for all ${\displaystyle \lambda \in \mathbb {D} }$.

Georg Pick and Rolf Nevanlinna solved the problem independently in 1916 and 1919 respectively, showing that an interpolating function exists if and only if a matrix defined in terms of the initial and target data is positive semi-definite.