Interpolation by Bloch functions
type: Item
instance of: scholarly article
title: Interpolation by Bloch functions
zbMATH Open document ID: 1053.30022
author: Alexander P. Schuster
published in: Illinois Journal of Mathematics
publication date: 1999-01-01T00:00:00Z
review text: Let \(\Gamma = \{z_ n\}\) be an infinite sequence of distinct points in the unit disk \(\mathbb D = \{z: | z| < 1\}\) having no accumulation points in \(\mathbb D.\) To this sequence, the space \(\ell^\infty(\Gamma),\) which consists of all sequences \(\{a_ n\}\) having the property that \(\sup_ n| a_ n|/\log1/(1-| z_ n|^2)<\infty,\) is associated. Given \(\Gamma,\) consider the linear operator \(T,\) which maps an analytic function \(f\) in \(\mathbb D\) to the sequence \(\{f(z_ n)\}_ n.\) A sequence \(\Gamma\) is said to be a set of interpolation for the Bloch space \(\mathbb B\) if \(T\) maps \(\mathbb B\) onto \(\ell^\infty(\Gamma).\) The aim of this paper is to find a complete geometric characterization of these sets. The author obtains some necessary and sufficient conditions for \(\Gamma\) to be a set of interpolation, as well as discovers some important properties of these sets. The sampling problem, in some sense the dual problem to interpolation, is examined. The natural question about sets of sampling concerns their existence. It turns out that there are no sets of sampling for the Bloch space, and therefore none for the little Bloch space either.
Mathematics Subject Classification ID: 30D45
zbMATH DE Number: 1361052
zbMATH Keywords: Bloch space
zbMATH Keywords: little Bloch space
zbMATH Keywords: Bergman space
zbMATH Keywords: set of interpolation
zbMATH Keywords: sampling problem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1361052