Incomplex analysis, giveninitial data consisting of${\displaystyle n}$ points${\displaystyle {\lambda }_1,\dots ,{\lambda }_n}$ in the complex unit disc${\displaystyle \mathbb{D}}$ andtarget data consisting of${\displaystyle n}$ points${\displaystyle z_1,\dots ,z_n}$ in${\displaystyle \mathbb{D}}$, theNevanlinna–Pick interpolation problem is to find aholomorphic function${\displaystyle \phi }$ thatinterpolates the data, that is for all${\displaystyle i\in \{1,...,n\}}$,

${\displaystyle \phi ({\lambda }_i)=z_i}$,

subject to the constraint${\displaystyle \left|\phi (\lambda )\right|\le 1}$ for all${\displaystyle \lambda \in \mathbb{D}}$.

Georg Pick andRolf 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 ispositive semi-definite.

The Nevanlinna–Pick theorem represents an${\displaystyle n}$-point generalization of theSchwarz lemma. Theinvariant form of the Schwarz lemma states that for a holomorphic function${\displaystyle f:\mathbb{D}\to \mathbb{D}}$, for all${\displaystyle {\lambda }_1,{\lambda }_2\in \mathbb{D}}$,

${\displaystyle \left|\frac{f({\lambda }_1)-f({\lambda }_2)}{1-\overline{f({\lambda }_2)}f({\lambda }_1)}\right|\le \left|\frac{{\lambda }_1-{\lambda }_2}{1-\overline{{\lambda }_2}{\lambda }_1}\right|.}$

Setting${\displaystyle f({\lambda }_i)=z_i}$, this inequality is equivalent to the statement that the matrix given by

that is thePick matrix is positive semidefinite.

Combined with the Schwarz lemma, this leads to the observation that for${\displaystyle {\lambda }_1,{\lambda }_2,z_1,z_2\in \mathbb{D}}$, there exists a holomorphic function${\displaystyle \phi :\mathbb{D}\to \mathbb{D}}$ such that${\displaystyle \phi ({\lambda }_1)=z_1}$ and${\displaystyle \phi ({\lambda }_2)=z_2}$ if and only if the Pick matrix

${\displaystyle {\left(\frac{1-\overline{z_j}{z}_i}{1-\overline{{\lambda }_j}{\lambda }_i}\right)}_{i,j=1,2}\ge 0.}$

The Nevanlinna–Pick theorem states the following. Given${\displaystyle {\lambda }_1,\dots ,{\lambda }_n,z_1,\dots ,z_n\in \mathbb{D}}$, there exists a holomorphic function${\displaystyle \phi :\mathbb{D}\to \overline{\mathbb{D}}}$ such that${\displaystyle \phi ({\lambda }_i)=z_i}$ if and only if the Pick matrix

${\displaystyle {\left(\frac{1-\overline{z_j}{z}_i}{1-\overline{{\lambda }_j}{\lambda }_i}\right)}_{i,j=1}^n}$

is positive semi-definite. Furthermore, the function${\displaystyle \phi }$ is unique if and only if the Pick matrix has zerodeterminant. In this case,${\displaystyle \phi }$ is aBlaschke product, with degree equal to the rank of the Pick matrix (except in the trivial case whereall the${\displaystyle z_i}$'s are the same).

The generalization of the Nevanlinna–Pick theorem became an area of active research inoperator theory following the work ofDonald Sarason on theSarason interpolation theorem.^[1] Sarason gave a new proof of the Nevanlinna–Pick theorem usingHilbert space methods in terms ofoperator contractions. Other approaches were developed in the work ofL. de Branges, andB. Sz.-Nagy andC. Foias.

It can be shown that theHardy spaceH ² is areproducing kernel Hilbert space, and that its reproducing kernel (known as theSzegő kernel) is

${\displaystyle K(a,b)={\left(1-b\overline{a}\right)}^{-1}.}$

Because of this, the Pick matrix can be rewritten as

${\displaystyle {\left((1-z_i\overline{z_j})K({\lambda }_j,{\lambda }_i)\right)}_{i,j=1}^N.}$

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${\displaystyle f:R\to \mathbb{D}}$ that interpolates a given set of data, whereR is now an arbitrary region of the complex plane.

M. B. Abrahamse showed that if the boundary ofR consists of finitely many analytic curves (sayn + 1), then an interpolating functionf exists if and only if

${\displaystyle {\left((1-z_i\overline{z_j})K_{\tau }({\lambda }_j,{\lambda }_i)\right)}_{i,j=1}^N}$

is a positive semi-definite matrix, for all${\displaystyle \tau }$ in then-torus. Here, the${\displaystyle K_{\tau }}$s are the reproducing kernels corresponding to a particular set of reproducing kernel Hilbert spaces, which are related to the setR. It can also be shown thatf is unique if and only if one of the Pick matrices has zero determinant.

• Pick's original proof concerned functions with positive real part. Under a linear fractionalCayley transform, his result holds on maps from the disc to the disc.

• Pick–Nevanlinna interpolation was introduced intorobust control byAllen Tannenbaum.

• The Pick-Nevanlinna problem for holomorphic maps from the bidisk${\displaystyle {\mathbb{D}}^2}$ to the disk was solved byJim Agler.

• Agler, Jim; John E. McCarthy (2002).Pick Interpolation and Hilbert Function Spaces.Graduate Studies in Mathematics.AMS.ISBN 0-8218-2898-3.
• Abrahamse, M. B. (1979)."The Pick interpolation theorem for finitely connected domains".Michigan Math. J.26 (2):195–203.doi:10.1307/mmj/1029002212.
• Tannenbaum, Allen (1980). "Feedback stabilization of linear dynamical plants with uncertainty in the gain factor".Int. J. Control.32 (1):1–16.doi:10.1080/00207178008922838.

• Interpolation