Nevanlinna–Pick interpolation explained

In complex analysis, given initial data consisting of

n points λ _{1,} \ldots, λ _{n} in the complex unit disc D and target data consisting of n points z _{1,} \ldots, z _{n} in 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( λ _{i)} = z _{i} ,subject to the constraint \left\vert \varphi( λ ) \right\vert \le 1 for all λ \in 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: D \to D , for all λ _{1,} λ _{2} \in D ,\left| (f( λ _{1)} - f( λ _{2)})/(1- \overline{f( λ _{2)}) f( λ _{1)}\right|} \leq \left| (λ _{1} -λ _{2})/(1- \overline{ λ _{2}) λ _{1}\right|.}

Setting

f( λ _{i)=z} _{i} , this inequality is equivalent to the statement that the matrix given by\begin{bmatrix} (1 - |z ^{2} _{1 |})/(1 - | λ ^{2} _{1 |}) & (1- \overline{z _{1})/(z) _{2}{1} - \overline{ λ _{1}λ} _{2} \\[5pt]} (1- \overline{z _{2})/(z) _{1}{1} - \overline{ λ _{2}λ} _{1}} & (1 - |z ^{2} _{2 |})/(1 - | λ ^{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

λ _{1,} λ _{2,} z _{1,} z _{2} \in D , there exists a holomorphic function \varphi: D \to D such that \varphi( λ _{1)} = z _{1} and \varphi( λ _{2)=z} _{2} if and only if the Pick matrix\left( (1- \overline{z _{j})/(z) _{i}{1} - \overline{ λ _{j}λ} _{i}\right)} _{i,j} \geq 0 .

The Nevanlinna–Pick theorem

The Nevanlinna–Pick theorem states the following. Given

λ _{1,} \ldots, λ _{n,} z _{1,} \ldots, z _{n} \in D , there exists a holomorphic function \varphi: D \to \overline{ D } such that \varphi( λ _{i)} = z _{i} if and only if the Pick matrix\left( (1- \overline{z _{j})/(z) _{i}{1-\overline{λ} _{j}} λ _{i}} \right) ^{n} _{i,j =1}

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 whereall the z _{i} 's are the same).

Generalisation

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.^[1] 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( λ _{j,λ} _{i)\right)} ^{N.} ^{i,j =1}

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 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} ( λ _{j,λ} _{i)\right)} ^{N} ^{i,j =1}

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.

Notes

• Pick's original proof concerned functions with positive real part. Under a linear fractional Cayley transform, his result holds on maps from the disc to the disc.
• Pick–Nevanlinna interpolation was introduced into robust control by Allen Tannenbaum.
• The Pick-Nevanlinna problem for holomorphic maps from the bidisk

D ^{2} to the disk was solved by Jim Agler.

References

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

Notes and References

1. Sarason. Donald. Generalized Interpolation in H ^{infty} . Trans. Amer. Math. Soc.. 1967. 127. 179–203. 10.1090/s0002-9947-1967-0208383-8. free.