Radial limits of interpolating Blaschke products
type: Item
instance of: scholarly article
title: Radial limits of interpolating Blaschke products
zbMATH Open document ID: 1068.30026
author: Pamela Gorkin
author: Raymond Mortini
published in: Mathematische Annalen
publication date: 2005-04-15T00:00:00Z
review text: Let \(B\) be a Blaschke product with the zero sequence \((z_n)_n\subset\mathbb D=\) the unit disc. We say that \(B\) is an interpolating Blaschke product if for any bounded sequence \((w_n)_n\subset\mathbb C\) there exists a bounded holomorphic function \(f\) in \(\mathbb D\) with \(f(z_n)=w_n\) for all \(n\). The authors investigate the following interpolation problem. Given points \((\lambda_n)_n\subset\partial\mathbb D\), \((a_n)_n\subset\overline{\mathbb D}\) find conditions under which there exists an interpolating Blaschke product \(B\) such that for each \(n\) the radial limit of \(B\) at \(\lambda_n\) equals \(a_n\). The main result of the paper says that if \((\lambda_n)_n\subset \partial\mathbb D\), then the problem has a solution iff \(\inf_n| a_n| >0\). See also: [\textit{G. Cargo}, J. Math. Anal. Appl. 71, 287-296 (1979; Zbl 0428.30021); \textit{C. Belna, F. Carroll, G. Piranian}, Trans. Am. Math. Soc. 280, 695--702 (1983; Zbl 0549.30023); \textit{C. Belna, P. Colwell, G. Piranian}, Proc. Am. Math. Soc. 93, 267--271 (1985; Zbl 0582.30022)].
reviewed by: Marek Jarnicki
Mathematics Subject Classification ID: 30D50
zbMATH DE Number: 2156227
MaRDI profile type: Publication
full work available at URL: s00208-004-0588-0
OpenAlex ID: W1975302796
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DOI: 10.1007/S00208-004-0588-0
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description: scientific article; zbMATH DE number 2156227