Nevanlinna-Pick interpolation
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In complex analysis, Nevanlinna-Pick interpolation is the problem of finding a holomorphic function from the unit disc to the unit disc (denoted ), which takes specified points to specified points. Equivalently, it is the problem of finding a holomorphic function f which interpolates a data set, subject to the upper bound for all .
More formally, if z1, ..., zN and w1, ..., wN are collections of points in the unit disc, the Nevanlinna-Pick problem is the problem of finding a holomorphic function
such that
- f(wi) = zi for all i between 1 and N
The problem was independently solved by G. Pick and R. Nevanlinna in 1916 and 1919 respectively. It was shown that such an f exists if and only if the Pick matrix
is positive semi-definite. Also, the function f is unique if and only if the Pick matrix has zero determinant. Pick's original proof was based on Blaschke products.
[edit] Generalisation
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
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.
[edit] References
- 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.