In
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
, 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
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
that
interpolate
In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.
In engineering and science, one often has a n ...
s the data, that is for all
,
:
,
subject to the constraint
for all
.
Georg Pick
Georg Alexander Pick (10 August 1859 – 26 July 1942) was an Austrian Jewish mathematician who was murdered during The Holocaust. He was born in Vienna to Josefa Schleisinger and Adolf Josef Pick and died at Theresienstadt concentration camp. Toda ...
and
Rolf Nevanlinna
Rolf Herman Nevanlinna (né Neovius; 22 October 1895 – 28 May 1980) was a Finnish mathematician who made significant contributions to complex analysis.
Background
Nevanlinna was born Rolf Herman Neovius, becoming Nevanlinna in 1906 when his fa ...
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
-point generalization of the
Schwarz lemma
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapping ...
. 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
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 mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...
. In this case,
is a
Blaschke product
In complex analysis, the Blaschke product is a bounded analytic function in the open unit disc constructed to have zeros at a (finite or infinite) sequence of prescribed complex numbers
:''a''0, ''a''1, ...
inside the unit disc, with the property ...
, with degree equal to the rank of the Pick matrix (except in the trivial case where
all the
's are the same).
Generalisation
The generalization of the Nevanlinna–Pick theorem became an area of active research in
operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear oper ...
following the work of
Donald Sarason
Donald Erik Sarason (January 26, 1933 – April 8, 2017) was an American mathematician who made fundamental advances in the areas of Hardy space theory and VMO. He was one of the most popular doctoral advisors in the Mathematics Department at U ...
on the
Sarason interpolation theorem.
Sarason gave a new proof of the Nevanlinna–Pick theorem using
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
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
In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . In ...
''H''
2 is a
reproducing kernel Hilbert space
In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Roughly speaking, this means that if two functions f and g i ...
, 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
in the
''n''-torus. Here, the
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 In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform i ...
, his result holds on maps from the disc to the disc.
* Pick–Nevanlinna interpolation was introduced into
robust control In control theory, robust control is an approach to controller design that explicitly deals with uncertainty. Robust control methods are designed to function properly provided that uncertain parameters or disturbances are found within some (typicall ...
by
Allen Tannenbaum
Allen Robert Tannenbaum (born January 25, 1953) is an American/ Israeli applied mathematician and presently Distinguished Professor of Computer Science and Applied Mathematics & Statistics at the State University of New York at Stony Brook. He is ...
.
References
*
*
*
{{DEFAULTSORT:Nevanlinna-Pick interpolation
Interpolation