# Nevanlinna–Pick interpolation

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

## Background

The Nevanlinna–Pick theorem represents an $n$-point generalization of the Schwarz lemma. The invariant form of the Schwarz lemma states that for a holomorphic function $f:\mathbb {D} \to \mathbb {D}$, for all $\lambda _{1},\lambda _{2}\in \mathbb {D}$,
    $\left|{\frac {f(\lambda _{1})-f(\lambda _{2})}{1-{\overline {f(\lambda _{2})}}f(\lambda _{1})}}\right|\leq \left|{\frac {\lambda _{1}-\lambda _{2}}{1-{\overline {\lambda _{2}}}\lambda _{1}}}\right|.$
Setting $f(\lambda _{i})=z_{i}$, this inequality is equivalent to the statement that the matrix given by
    ${\begin{bmatrix}{\frac {1-|z_{1}|^{2}}{1-|\lambda _{1}|^{2}}}&{\frac {1-{\overline {z_{1}}}z_{2}}{1-{\overline {\lambda _{1}}}\lambda _{2}}}\\[5pt]{\frac {1-{\overline {z_{2}}}z_{1}}{1-{\overline {\lambda _{2}}}\lambda _{1}}}&{\frac {1-|z_{2}|^{2}}{1-|\lambda _{2}|^{2}}}\end{bmatrix}}\geq 0,$
that is the Pick matrix is positive semidefinite.
Combined with the Schwarz lemma, this leads to the observation that for $\lambda _{1},\lambda _{2},z_{1},z_{2}\in \mathbb {D}$, there exists a holomorphic function $\varphi :\mathbb {D} \to \mathbb {D}$ such that $\varphi (\lambda _{1})=z_{1}$ and $\varphi (\lambda _{2})=z_{2}$ if and only if the Pick matrix
    $\left({\frac {1-{\overline {z_{j}}}z_{i}}{1-{\overline {\lambda _{j}}}\lambda _{i}}}\right)_{i,j=1,2}\geq 0.$

## The Nevanlinna–Pick theorem

The Nevanlinna–Pick theorem states the following. Given $\lambda _{1},\ldots ,\lambda _{n},z_{1},\ldots ,z_{n}\in \mathbb {D}$, there exists a holomorphic function $\varphi :\mathbb {D} \to {\overline {\mathbb {D} }}$ such that $\varphi (\lambda _{i})=z_{i}$ if and only if the Pick matrix
    $\left({\frac {1-{\overline {z_{j}}}z_{i}}{1-{\overline {\lambda _{j}}}\lambda _{i}}}\right)_{i,j=1}^{n}$
is positive semi-definite. Furthermore, the function $\varphi$ is unique if and only if the Pick matrix has zero determinant. In this case, $\varphi$ is a Blaschke product, with degree equal to the rank of the Pick matrix (except in the trivial case where all the $z_{i}$'s are the same).

## Generalization

The generalization of the Nevanlinna–Pick theorem became an area of active research in operator theory following the work of Donald Sarason on the Sarason interpolation theorem. Sarason gave a new proof of the Nevanlinna–Pick theorem using Hilbert space methods in terms of operator contractions. Other approaches were developed in the work of L. de Branges, and B. Sz.-Nagy and C. Foias.
It can be shown that the Hardy space H$^2$ is a reproducing kernel Hilbert space, and that its reproducing kernel (known as the Szegő kernel) is
    $K(a,b)=\left(1-b{\bar {a}}\right)^{-1}.\,$
Because of this, the Pick matrix can be rewritten as
    $\left((1-z_{i}{\overline {z_{j}}})K(\lambda _{j},\lambda _{i})\right)_{i,j=1}^{N}.\,$
This description of the solution has motivated various attempts to generalise Nevanlinna and Pick's result.
The Nevanlinna–Pick problem can be generalised to that of finding a holomorphic function $f:R\to \mathbb {D}$ that interpolates a given set of data, where R is now an arbitrary region of the complex plane.
M. B. Abrahamse showed that if the boundary of R consists of finitely many analytic curves (say n + 1), then an interpolating function f exists if and only if
    $\left((1-z_{i}{\overline {z_{j}}})K_{\tau }(\lambda _{j},\lambda _{i})\right)_{i,j=1}^{N}\,$
is a positive semi-definite matrix, for all $\tau$ in the n-torus. Here, the $K_{\tau }$s are the reproducing kernels corresponding to a particular set of reproducing kernel Hilbert spaces, which are related to the set R. It can also be shown that f is unique if and only if one of the Pick matrices has zero determinant.