How to get universal inner functions
type: Item
instance of: scholarly article
title: How to get universal inner functions
zbMATH Open document ID: 1129.47027
author: Pamela Gorkin
author: Frédéric Bayart
published in: Mathematische Annalen
publication date: 2007-09-11T00:00:00Z
review text: Many interesting results concerning universal inner functions have been proved recently by R.\,Aron, R.\,Mortini and the authors of this paper. Their purpose here is to present a result which puts the existence of universal inner functions in a more general framework. More precisely, a condition, called the mutual approximation property of the triple \((U,\mathcal{B},\mathcal{I})\) of a domain \(U\) in \(\mathbb{C}^d\), and subsets \(\mathcal{B}\) and \(\mathcal{I}\) of the closed unit ball of \(H^{\infty}(U)\), is introduced. The main theorem asserts that if \(\mathcal{I}\) is closed under multiplication and automorphisms, the constant function \(1\) belongs to both \(\mathcal{B}\) and \(\mathcal{I}\), and \((U,\mathcal{B},\mathcal{I})\) satisfies the mutual approximation property, then there is a closed infinite-dimensional vector space, generated by functions in \(\mathcal{I}\), isometric to \(\ell_1\), such that all its elements are universal. The assumptions are satisfied by the set \(\mathcal{I}\) of Blaschke products on the unit disc in \(\mathbb{C}\) and by the set of inner functions on the ball or, more generally, on a bounded symmetric domain. Therefore, the authors obtain as a consequence all known results and many new ones.
reviewed by: José Bonet
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 46J10
Mathematics Subject Classification ID: 32A38
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 5189572
zbMATH Keywords: universal functions
zbMATH Keywords: inner functions
zbMATH Keywords: mutual approximation property
zbMATH Keywords: Blaschke product
MaRDI profile type: Publication
full work available at URL: s00208-006-0059-x
OpenAlex ID: W2004560324
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DOI: 10.1007/S00208-006-0059-X
description: scientific article; zbMATH DE number 5189572