Larson-Miller Parameter

In complex analysis, Nevanlinna–Pick interpolation is the problem of finding a holomorphic function from the unit disc to the unit disc (denoted $\mathbb{𝔻}$), 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 $|f(z)|\le 1$ for all $z\in \mathbb{𝔻}$.

More formally, if z₁, ..., z_N and w₁, ..., w_N are collections of points in the unit disc, the Nevanlinna–Pick problem is the problem of finding a holomorphic function

$f:\mathbb{𝔻}\to \mathbb{𝔻}$

such that

f(z_i) = w_i 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

${\left(\frac{1-w_i\overline{w_j}}{1-z_i\overline{z_j}}\right)}_{i,j=1}^N$

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. Pick-Nevanlinna interpolation was introduced into robust control by Allen Tannenbaum.

The Cayley transform can be used to to solve the Pick–Nevanlinna problem for the complex half plane.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

Generalisation

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

$K(a,b)={(1-b\overline{a})}^{-1}.$

Because of this, the Pick matrix can be rewritten as

${((1-w_i\overline{w_j})K(z_j,z_i))}_{i,j=1}^N.$

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 $f:R\to \mathbb{𝔻}$ 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

${((1-w_i\overline{w_j})K_{\lambda }(z_j,z_i))}_{i,j=1}^N$

is a positive semi-definite matrix, for all λ in the n-torus. Here, the K_λs 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

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