Thin interpolating sequences in the disk
type: Item
instance of: scholarly article
title: Thin interpolating sequences in the disk
zbMATH Open document ID: 1179.30058
author: Raymond Mortini
published in: Archiv der Mathematik
publication date: 2009-06-16T00:00:00Z
review text: The author gives a new proof of an important result of \textit{C. Sundberg} and \textit{T. H. Wolff} [Trans. Am. Math. Soc. 276, 551--581 (1983; Zbl 0536.30025)]: Let \((z_n)\) be a thin interpolating sequence in \(\mathbb{D}=\{|z|<1\}\). Then there are numbers \(\tau_j\in]0,1[\) and \(\gamma_j\in]0,1[\) with \(\lim\tau_j=\lim \lambda_j=1\) such that, for every sequence \((\xi_j)^n_1\) in \(\mathbb{D}\) with \(|(z_j-\xi_j)/(1-\overline z_j\xi_j)|<\tau_j\), then \(\prod_{k:k\neq j}|(\xi_j-\xi_k)/(1-\overline \xi_j\xi_k)|\geq\gamma_j\). He also proves that every asymptotic interpolation problem \(|f(a_n)-w_n|\to 0\), \(\sup_n|w_n|\leq 1\), can be solved by a thin Blaschke product whenever \((a_n)\) is thin. This result appeared implicitly earlier in work of \textit{K. Dyakonov} and \textit{A. Nicolau} [Trans. Am. Math. Soc. 359, No. 9, 4449--4465 (2007; Zbl 1141.30010)].
Mathematics Subject Classification ID: 30J10
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 5565264
zbMATH Keywords: thin interpolating sequences
zbMATH Keywords: asymptotic interpolation
reviewed by: David A. Brannan
MaRDI profile type: Publication
full work available at URL: s00013-009-3057-x
OpenAlex ID: W2060820415
DOI: 10.1007/S00013-009-3057-X
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description: scientific article; zbMATH DE number 5565264