Nevanlinna–Pick interpolation

In complex analysis, given initial data consisting of ${\displaystyle n}$ points ${\displaystyle \lambda _{1},\ldots ,\lambda _{n}}$ in the complex unit disc ${\displaystyle \mathbb {D} }$ and target data consisting of ${\displaystyle n}$ points ${\displaystyle z_{1},\ldots ,z_{n}}$ in ${\displaystyle \mathbb {D} }$, the Nevanlinna–Pick interpolation problem is to find a holomorphic function ${\displaystyle \varphi }$ that interpolates the data, that is for all ${\displaystyle i}$,

${\displaystyle \varphi (\lambda _{i})=z_{i}}$,

subject to the constraint ${\displaystyle \left\vert \varphi (\lambda )\right\vert \leq 1}$ for all ${\displaystyle \lambda \in \mathbb {D} }$.

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 ${\displaystyle n}$-point generalization of the Schwarz lemma. The invariant form of the Schwarz lemma states that for a holomorphic function ${\displaystyle f:\mathbb {D} \to \mathbb {D} }$, for all ${\displaystyle \lambda _{1},\lambda _{2}\in \mathbb {D} }$,

${\displaystyle \left|{\frac {f(\lambda _{1})-f(\lambda _{2})}{1-{\overline {f(\lambda _{2})}}f(\lambda _{1})}}\right|\leq \left|{\frac {\lambda _{1}-\lambda _{2}}{1-{\overline {\lambda _{2}}}\lambda _{1}}}\right|.}$

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

${\displaystyle {\begin{bmatrix}{\frac {1-|z_{1}|^{2}}{1-|\lambda _{1}|^{2}}}&{\frac {1-{\overline {z_{1}}}z_{2}}{1-{\overline {\lambda _{1}}}\lambda _{2}}}\\[5pt]{\frac {1-{\overline {z_{2}}}z_{1}}{1-{\overline {\lambda _{2}}}\lambda _{1}}}&{\frac {1-|z_{2}|^{2}}{1-|\lambda _{2}|^{2}}}\end{bmatrix}}\geq 0,}$

that is the Pick matrix is positive semidefinite.

Combined with the Schwarz lemma, this leads to the observation that for ${\displaystyle \lambda _{1},\lambda _{2},z_{1},z_{2}\in \mathbb {D} }$, there exists a holomorphic function ${\displaystyle \varphi$ :\mathbb {D} \to \mathbb {D} } such that ${\displaystyle \varphi (\lambda _{1})=z_{1}}$ and ${\displaystyle \varphi (\lambda _{2})=z_{2}}$ if and only if the Pick matrix

${\displaystyle \left({\frac {1-{\overline {z_{j}}}z_{i}}{1-{\overline {\lambda _{j}}}\lambda _{i}}}\right)_{i,j=1,2}\geq 0.}$

The Nevanlinna–Pick theorem

The Nevanlinna–Pick theorem states the following. Given ${\displaystyle \lambda _{1},\ldots ,\lambda _{n},z_{1},\ldots ,z_{n}\in \mathbb {D} }$, there exists a holomorphic function ${\displaystyle \varphi$ :\mathbb {D} \to {\overline {\mathbb {D} }}} such that ${\displaystyle \varphi (\lambda _{i})=z_{i}}$ 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 ${\displaystyle \varphi }$ is unique if and only if the Pick matrix has zero determinant. In this case, ${\displaystyle \varphi }$ is a Blaschke product, with degree equal to the rank of the Pick matrix (except in the trivial case where all the ${\displaystyle z_{i}}$'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

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

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, 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

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

is a positive semi-definite matrix, for all ${\displaystyle \tau }$ in the n-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 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

1. Sarason, Donald (1967). "Generalized Interpolation in ${\displaystyle H^{\infty }}$". 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. J. Control. 32 (1): 1–16. doi:10.1080/00207178008922838.