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 $i$ ,

\varphi(\lambda_i) = z_i

subject to the constraint $\left\vert \varphi(\lambda) \right\vert \le 1$ 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} \\ \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.

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 H² 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-w_i \overline{w_j}) K(z_j,z_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-w_i \overline{w_j}) K_\lambda (z_j,z_i)\right)_{i,j=1}^N\,

is a positive semi-definite matrix, for all λ in the n-torus. Here, the K_λs 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.

References

↑ Sarason, Donald (1967). "Generalized Interpolation in $H^\infty$ ". Trans. Amer. Math. Soc. 127: 179–203. doi:10.1090/s0002-9947-1967-0208383-8.

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. Journal Control 32 (1): 1–16. doi:10.1080/00207178008922838.