Multi-point degenerate interpolation problem for generalized Schur functions: description of all solutions
type: Item
instance of: scholarly article
title: Multi-point degenerate interpolation problem for generalized Schur functions: description of all solutions
zbMATH Open document ID: 1246.30068
DOI: 10.1007/BF03321794
author: Vladimir Bolotnikov
published in: Computational Methods and Function Theory
publication date: 2011-10-28T00:00:00Z
review text: The generalized Schur class \(S_k\) is defined as the set of meromorphic functions in the unit disk of the form \[ f(z)=\frac{s(z)}{b(z)}, \] where \(s\) is a Schur function and \(b\) is a Blaschke product of degree \(k\). The author studies the general Nevanlinna-Pick type interpolation problem in the class \(S_k\): Given the data set \[ \{z_i,\;n_i,\;c_{ij}: \;j=0,1,\dots,n_i-1; \;i=1,2,\dots,k\}, \] where \(\{z_1,\dots,z_k\}\) are distinct points in the disk, \((n_1,\dots,n_k)\) is a tuple of multiplicities, \(\{c_{ij}\}\) are complex numbers, find all solutions \(f\in S_k\), analytic at \(z_i\) and \[ \frac{f^{(j)}(z_i)}{j!}=c_{ij}. \] The objective of the paper is to present an algorithm providing all solutions to this problem in the degenerate indeterminate case. The algorithm suggested in the paper works equally well for every \(k\) for which the problem has infinitely many solutions.
Mathematics Subject Classification ID: 30E05
zbMATH DE Number: 5965032
zbMATH Keywords: generalized Schur functions
zbMATH Keywords: Nevanlinna-Pick interpolation problem
zbMATH Keywords: Pick matrix
zbMATH Keywords: linear fractional transformation
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5965032