Nevanlinna–Pick interpolation

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 $f:\backslash mathbb\{D\}\backslash to\backslash mathbb\{D\}$, for all $\backslash lambda\_1,\; \backslash lambda\_2\; \backslash in\; \backslash mathbb\{D\}$,

:$\backslash left|\backslash frac\{f(\backslash lambda\_1)\; -\; f(\backslash lambda\_2)\}\{1\; -\; \backslash overline\{f(\backslash lambda\_2)\}f(\backslash lambda\_1)\}\backslash right|\; \backslash leq\; \backslash left|\backslash frac\{\backslash lambda\_1\; -\; \backslash lambda\_2\}\{1\; -\; \backslash overline\{\backslash lambda\_2\}\backslash lambda\_1\}\backslash right|.$

Setting $f(\backslash lambda\_i)=z\_i$, this inequality is equivalent to the statement that the matrix given by

:

\\[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 $\backslash lambda\_1,\; \backslash lambda\_2,\; z\_1,\; z\_2\; \backslash in\; \backslash mathbb\{D\}$, there exists a holomorphic function $\backslash varphi:\backslash mathbb\{D\}\; \backslash to\; \backslash mathbb\{D\}$ such that $\backslash varphi(\backslash lambda\_1)\; =\; z\_1$ and $\backslash varphi(\backslash lambda\_2)=z\_2$ if and only if the Pick matrix

:$\backslash left(\backslash frac\{1\; -\; \backslash overline\{z\_j\}z\_i\}\{1\; -\; \backslash overline\{\backslash lambda\_j\}\backslash lambda\_i\}\backslash right)\_\{i,j\; =\; 1,\; 2\}\; \backslash geq\; 0.$

The Nevanlinna–Pick theorem states the following. Given $\backslash lambda\_1,\; \backslash ldots,\; \backslash lambda\_n,\; z\_1,\; \backslash ldots,\; z\_n\; \backslash in\; \backslash mathbb\{D\}$, there exists a holomorphic function $\backslash varphi:\backslash mathbb\{D\}\; \backslash to\; \backslash overline\{\backslash mathbb\{D\}\}$ such that $\backslash varphi(\backslash lambda\_i)\; =\; z\_i$ if and only if the Pick matrix

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

is positive semi-definite. Furthermore, the function $\backslash varphi$ is unique if and only if the Pick matrix has zero determinant. In this case, $\backslash varphi$ 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).

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.{{cite journal|last=Sarason|first=Donald|title=Generalized Interpolation in $H^\backslash infty$|journal=Trans. Amer. Math. Soc.|date=1967|volume=127|pages=179–203|doi=10.1090/s0002-9947-1967-0208383-8|doi-access=free}} 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

:$K(a,b)=\backslash left(1-b\; \backslash bar\{a\}\; \backslash right)^\{-1\}.\backslash ,$

Because of this, the Pick matrix can be rewritten as

:$\backslash left(\; (1-z\_i\; \backslash overline\{z\_j\})\; K(\backslash lambda\_j,\backslash lambda\_i)\backslash right)\_\{i,j=1\}^N.\backslash ,$

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 $f:R\backslash to\backslash mathbb\{D\}$ 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

:$\backslash left(\; (1-z\_i\; \backslash overline\{z\_j\})\; K\_\backslash tau\; (\backslash lambda\_j,\backslash lambda\_i)\backslash right)\_\{i,j=1\}^N\backslash ,$

is a positive semi-definite matrix, for all $\backslash tau$ in the n-torus. Here, the $K\_\backslash tau$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.

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

• Pick–Nevanlinna interpolation was introduced into robust control by Allen Tannenbaum.

• The Pick–Nevanlinna problem for holomorphic maps from the bidisk $\backslash mathbb\{D\}^2$ to the disk was solved by Jim Agler.

{{reflist}}

• {{cite book

| last = Agler

| first = Jim

|author2=John E. McCarthy

| title = Pick Interpolation and Hilbert Function Spaces

| publisher = AMS

| series = Graduate Studies in Mathematics

| year = 2002

| isbn = 0-8218-2898-3 }}

• {{cite journal

| first = M. B.

| last = Abrahamse

| title = The Pick interpolation theorem for finitely connected domains

| year = 1979

| journal = Michigan Math. J.

| volume = 26

| issue = 2

| pages = 195–203

| doi = 10.1307/mmj/1029002212

| doi-access = free

}}

• {{cite journal

| first = Allen

| last = Tannenbaum

| title = Feedback stabilization of linear dynamical plants with uncertainty in the gain factor

| year = 1980

| journal = Int. J. Control

| volume = 32

| issue = 1

| pages = 1–16

| doi=10.1080/00207178008922838

}}

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Category:Interpolation