Boundary interpolation with Blaschke products of minimal degree
type: Item
instance of: scholarly article
title: Boundary interpolation with Blaschke products of minimal degree
zbMATH Open document ID: 1136.30010
DOI: 10.1007/BF03321626
author: G. Semmler
author: E. Wegert
published in: Computational Methods and Function Theory
publication date: 2007-05-14T00:00:00Z
review text: The classical Nevanlinna-Pick interpolation problem is to find functions \(f\), analytic in the unit disc \(\mathbb D\), which are bounded by \(1\) and satisfying the interpolation conditions \(f(z_i)=w_i\) (\(i=1,\dots,n\)) for given points \(z_1,\dots,z_n\) in the open unit disk \({\mathbb D}\) and complex numbers \(w_1,\dots,w_n\). Nevanlinna and Pick provided necessary and sufficient conditions for the solvability of the problem [see \textit{J. B. Garnett}, ``Bounded analytic functions.'' Grad. Texts Math. 236 (2006; Zbl 1106.30001)]. We remark that when the problem is solvable it has a solution which is a finite Blaschke product of degree at most \(n\). In this paper the author considers a natural modification of the Nevanlinna-Pick problem for Blaschke products. Namely, he places all the interpolation points \(z_k\) in the unit circle \(\mathbb T\). Since Blaschke products are unimodular on \(\mathbb T\), the interpolations values \(w_k\) must also lie on \(\mathbb T\). That is, the paper deals with the following boundary interpolation problem: Given \(n\) distinct points \(z_ 1,\dots,z_ n\in\mathbf C\) and \(n\) points \(w_ 1,\dots,w_ n\in\mathbf C\) with \(| z_ j| =| w_ j| =1, j=1,\dots, n\), find a finite Blaschke product \(B\) satisfying the interpolation condition \(B(z_ j)=w_ j, j=1,\dots,n\). \textit{D. G. Cantor} and \textit{R. R. Phelps} [Proc. Am. Math. Soc. 16, 523--525 (1965; Zbl 0146.09604)] proved that the problem is always solvable, \textit{R. Younis} [Proc. Am. Math. Soc. 78, No. 3, 451--452 (1980; Zbl 0439.30020)] proved that there is always a solution of degree at most \(n^2\) and \textit{W. B. Jones} and \textit{S. Ruscheweyh} [Constr. Approx. 3, No. 4, 405--409 (1987; Zbl 0634.41007)] improved this showing that there is always a solution of degree at most \(n-1\). The authors split the boundary interpolation problems into the following three classes depending on how the minimal possible degree \(m\) of a solution is related to the number \(…
reviewed by: Daniel Girela
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 30D50
zbMATH DE Number: 5152630
zbMATH Keywords: Boundary interpolation
zbMATH Keywords: Blaschke product
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5152630