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 $\left\vert \varphi (\lambda )\right\vert \leq 1$ 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} \\ \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.

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