Nevanlinna–Pick interpolation

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In complex analysis, given initial data consisting of points in the complex unit disc and target data consisting of points in , the Nevanlinna–Pick interpolation problem is to find a holomorphic function that interpolates the data, that is for all ,

,

subject to the constraint for all .

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[edit]

The Nevanlinna-Pick theorem represents an point generalization of the Schwarz lemma. The invariant form of the Schwarz lemma states that for a holomorphic function , for all ,

Setting , this inequality is equivalent to the statement that the matrix given by

that is the Pick matrix is positive semidefinite.

Combined with the Schwarz lemma, this leads to the observation that for , there exists a holomorphic function such that and if and only if the Pick matrix

The Nevanlinna-Pick Theorem[edit]

The Nevanlinna-Pick theorem states the following. Given , there exists a holomorphic function such that if and only if the Pick matrix

is positive semi-definite. Furthermore, the function is unique if and only if the Pick matrix has zero determinant. In this case, is a Blaschke product.

Generalisation[edit]

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

Because of this, the Pick matrix can be rewritten as

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 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

is a positive semi-definite matrix, for all λ in the n-torus. Here, the Kλs 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[edit]

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[edit]

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