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Nevanlinna–Pick interpolation

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In complex analysis, given initial data consisting of n points λ 1 , … , λ n in the complex unit disc D and target data consisting of n points z 1 , … , z n in D , the Nevanlinna–Pick interpolation problem is to find a holomorphic function φ that interpolates the data, that is for all i ,

Contents

φ ( λ i ) = z i ,

subject to the constraint | φ ( λ ) | ≤ 1 for all λ ∈ 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 → D , for all λ 1 , λ 2 ∈ D ,

| f ( λ 1 ) − f ( λ 2 ) 1 − f ( λ 2 ) ¯ f ( λ 1 ) | ≤ | λ 1 − λ 2 1 − λ 2 ¯ λ 1 | .

Setting f ( λ i ) = z i , this inequality is equivalent to the statement that the matrix given by

[ 1 − | z 1 | 2 1 − | λ 1 | 2 1 − z 1 ¯ z 2 1 − λ 1 ¯ λ 2 1 − z 2 ¯ z 1 1 − λ 2 ¯ λ 1 1 − | z 2 | 2 1 − | λ 2 | 2 ] ≥ 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 ∈ D , there exists a holomorphic function φ : D → D such that φ ( λ 1 ) = z 1 and φ ( λ 2 ) = z 2 if and only if the Pick matrix

( 1 − z j ¯ z i 1 − λ j ¯ λ i ) i , j = 1 , 2 ≥ 0.

The Nevanlinna-Pick Theorem

The Nevanlinna-Pick theorem states the following. Given λ 1 , … , λ n , z 1 , … , z n ∈ D , there exists a holomorphic function φ : D → D ¯ such that φ ( λ i ) = z i if and only if the Pick matrix

( 1 − z j ¯ z i 1 − λ j ¯ λ i ) i , j = 1 n

is positive semi-definite. Furthermore, the function φ is unique if and only if the Pick matrix has zero determinant. In this case, φ 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. 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 ) = ( 1 − b a ¯ ) − 1 .

Because of this, the Pick matrix can be rewritten as

( ( 1 − w i w j ¯ ) K ( z j , z i ) ) 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 → 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

( ( 1 − w i w j ¯ ) K λ ( z j , z i ) ) i , j = 1 N

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.

References

Nevanlinna–Pick interpolation Wikipedia
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