Nevanlinna–Pick interpolation

In complex analysis, given initial data consisting of ${\displaystyle n}${\displaystyle n} points ${\displaystyle \lambda _{1},\ldots ,\lambda _{n}}${\displaystyle \lambda _{1},\ldots ,\lambda _{n}} in the complex unit disk ${\displaystyle \mathbb {D} }${\displaystyle \mathbb {D} } and target data consisting of ${\displaystyle n}${\displaystyle n} points ${\displaystyle z_{1},\ldots ,z_{n}}${\displaystyle z_{1},\ldots ,z_{n}} in ${\displaystyle \mathbb {D} }${\displaystyle \mathbb {D} }, the Nevanlinna–Pick interpolation problem is to find a holomorphic function ${\displaystyle \varphi }${\displaystyle \varphi } that interpolates the data, that is for all ${\displaystyle i\in \{1,...,n\}}${\displaystyle i\in \{1,...,n\}},

${\displaystyle \varphi (\lambda _{i})=z_{i}}${\displaystyle \varphi (\lambda _{i})=z_{i}},

subject to the constraint ${\displaystyle \left\vert \varphi (\lambda )\right\vert \leq 1}${\displaystyle \left\vert \varphi (\lambda )\right\vert \leq 1} for all ${\displaystyle \lambda \in \mathbb {D} }${\displaystyle \lambda \in \mathbb {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.

The Nevanlinna–Pick theorem represents an ${\displaystyle n}${\displaystyle n}-point generalization of the Schwarz lemma. The invariant form of the Schwarz lemma states that for a holomorphic function ${\displaystyle f:\mathbb {D} \to \mathbb {D} }${\displaystyle f:\mathbb {D} \to \mathbb {D} }, for all ${\displaystyle \lambda _{1},\lambda _{2}\in \mathbb {D} }${\displaystyle \lambda _{1},\lambda _{2}\in \mathbb {D} },

${\displaystyle \left|{\frac {f(\lambda _{1})-f(\lambda _{2})}{1-{\overline {f(\lambda _{2})}}f(\lambda _{1})}}\right|\leq \left|{\frac {\lambda _{1}-\lambda _{2}}{1-{\overline {\lambda _{2}}}\lambda _{1}}}\right|.}${\displaystyle \left|{\frac {f(\lambda _{1})-f(\lambda _{2})}{1-{\overline {f(\lambda _{2})}}f(\lambda _{1})}}\right|\leq \left|{\frac {\lambda _{1}-\lambda _{2}}{1-{\overline {\lambda _{2}}}\lambda _{1}}}\right|.}

Setting ${\displaystyle f(\lambda _{i})=z_{i}}${\displaystyle f(\lambda _{i})=z_{i}}, this inequality is equivalent to the statement that the matrix given by

${\displaystyle {\begin{bmatrix}{\frac {1-|z_{1}|^{2}}{1-|\lambda _{1}|^{2}}}&{\frac {1-{\overline {z_{1}}}z_{2}}{1-{\overline {\lambda _{1}}}\lambda _{2}}}\\[5pt]{\frac {1-{\overline {z_{2}}}z_{1}}{1-{\overline {\lambda _{2}}}\lambda _{1}}}&{\frac {1-|z_{2}|^{2}}{1-|\lambda _{2}|^{2}}}\end{bmatrix}}\geq 0,}${\displaystyle {\begin{bmatrix}{\frac {1-|z_{1}|^{2}}{1-|\lambda _{1}|^{2}}}&{\frac {1-{\overline {z_{1}}}z_{2}}{1-{\overline {\lambda _{1}}}\lambda _{2}}}\\[5pt]{\frac {1-{\overline {z_{2}}}z_{1}}{1-{\overline {\lambda _{2}}}\lambda _{1}}}&{\frac {1-|z_{2}|^{2}}{1-|\lambda _{2}|^{2}}}\end{bmatrix}}\geq 0,}

that is the Pick matrix is positive semidefinite.

Combined with the Schwarz lemma, this leads to the observation that for ${\displaystyle \lambda _{1},\lambda _{2},z_{1},z_{2}\in \mathbb {D} }${\displaystyle \lambda _{1},\lambda _{2},z_{1},z_{2}\in \mathbb {D} }, there exists a holomorphic function ${\displaystyle \varphi :\mathbb {D} \to \mathbb {D} }${\displaystyle \varphi :\mathbb {D} \to \mathbb {D} } such that ${\displaystyle \varphi (\lambda _{1})=z_{1}}${\displaystyle \varphi (\lambda _{1})=z_{1}} and ${\displaystyle \varphi (\lambda _{2})=z_{2}}${\displaystyle \varphi (\lambda _{2})=z_{2}} if and only if the Pick matrix

${\displaystyle \left({\frac {1-{\overline {z_{j}}}z_{i}}{1-{\overline {\lambda _{j}}}\lambda _{i}}}\right)_{i,j=1,2}\geq 0.}${\displaystyle \left({\frac {1-{\overline {z_{j}}}z_{i}}{1-{\overline {\lambda _{j}}}\lambda _{i}}}\right)_{i,j=1,2}\geq 0.}

The Nevanlinna–Pick theorem states the following. Given ${\displaystyle \lambda _{1},\ldots ,\lambda _{n},z_{1},\ldots ,z_{n}\in \mathbb {D} }${\displaystyle \lambda _{1},\ldots ,\lambda _{n},z_{1},\ldots ,z_{n}\in \mathbb {D} }, there exists a holomorphic function ${\displaystyle \varphi :\mathbb {D} \to {\overline {\mathbb {D} }}}${\displaystyle \varphi :\mathbb {D} \to {\overline {\mathbb {D} }}} such that ${\displaystyle \varphi (\lambda _{i})=z_{i}}${\displaystyle \varphi (\lambda _{i})=z_{i}} if and only if the Pick matrix

${\displaystyle \left({\frac {1-{\overline {z_{j}}}z_{i}}{1-{\overline {\lambda _{j}}}\lambda _{i}}}\right)_{i,j=1}^{n}}${\displaystyle \left({\frac {1-{\overline {z_{j}}}z_{i}}{1-{\overline {\lambda _{j}}}\lambda _{i}}}\right)_{i,j=1}^{n}}

is positive semi-definite. Furthermore, the function ${\displaystyle \varphi }${\displaystyle \varphi } is unique if and only if the Pick matrix has zero determinant. In this case, ${\displaystyle \varphi }${\displaystyle \varphi } is a Blaschke product, with degree equal to the rank of the Pick matrix (except in the trivial case where all the ${\displaystyle z_{i}}${\displaystyle z_{i}}'s are the same).

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

${\displaystyle K(a,b)=\left(1-b{\bar {a}}\right)^{-1}.,円}${\displaystyle K(a,b)=\left(1-b{\bar {a}}\right)^{-1}.,円}

Because of this, the Pick matrix can be rewritten as

${\displaystyle \left((1-z_{i}{\overline {z_{j}}})K(\lambda _{j},\lambda _{i})\right)_{i,j=1}^{N}.,円}${\displaystyle \left((1-z_{i}{\overline {z_{j}}})K(\lambda _{j},\lambda _{i})\right)_{i,j=1}^{N}.,円}

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 ${\displaystyle f:R\to \mathbb {D} }${\displaystyle 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

${\displaystyle \left((1-z_{i}{\overline {z_{j}}})K_{\tau }(\lambda _{j},\lambda _{i})\right)_{i,j=1}^{N},円}${\displaystyle \left((1-z_{i}{\overline {z_{j}}})K_{\tau }(\lambda _{j},\lambda _{i})\right)_{i,j=1}^{N},円}

is a positive semi-definite matrix, for all ${\displaystyle \tau }${\displaystyle \tau } in the n-torus. Here, the ${\displaystyle K_{\tau }}${\displaystyle K_{\tau }}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.

• Pick's original proof concerned functions with positive real part. Under a linear fractional Cayley transform, his result holds on maps from the disk to the disk.

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

• The Pick–Nevanlinna problem for holomorphic maps from the bidisk ${\displaystyle \mathbb {D} ^{2}}${\displaystyle \mathbb {D} ^{2}} to the disk was solved by Jim Agler.

• 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. doi:10.1307/mmj/1029002212 .
• Tannenbaum, Allen (1980). "Feedback stabilization of linear dynamical plants with uncertainty in the gain factor". Int. J. Control. 32 (1): 1–16. doi:10.1080/00207178008922838.

• Interpolation

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