Interpolation by Lipschitz holomorphic functions
type: Item
instance of: scholarly article
title: Interpolation by Lipschitz holomorphic functions
zbMATH Open document ID: 0587.32009
DOI: 10.1007/BF02384432
author: Boguslaw Tomaszewski
published in: Arkiv för Matematik
publication date: 1985-01-01T00:00:00Z
review text: Let \(B=\{z\in {\mathbb{C}}^ d: | z| <1\}\) be the unit ball in \({\mathbb{C}}^ d\) for \(d>1\), \(S=\partial B\) and \(\sigma\) the Borel measure on S. The main theorem says that for every \(\epsilon >0\) there exists an \(\alpha\in (0,1]\) such that for every real function \(g\in Lip 1\) it is possible to find a nonconstant function \(f\in A(B)\cap Lip \alpha\) such that Re \(f\leq g\) and \(\sigma (\{z\in S:Re f(z)=g(z)\})\geq 1-\epsilon.\) (Here Lip \(\alpha\) is understood with respect to the length of any shortest path on S joining given points.) The following two corollaries are also derived: 1. For every \(\epsilon >0\) there exists an \(\alpha\in (0,1]\) such that for every positive function \(g\in Lip 1\) there exists a nonconstant function \(f\in A(B)\cap Lip \alpha\) such that \(| f| \leq g\) and \(\sigma (\{z\in S:| f(z)| =g(z)\})\geq 1-\epsilon;\) 2. There exists an \(\alpha\in (0,1]\) such that for every \(\epsilon >0\) it is possible to find a nonconstant function \(f\in A(B)\cap Lip \alpha\) such that \(\| f\|_{\infty}\leq 1\) and \(\sigma (\{z\in S: | f(z)| =1\})\geq 1-\epsilon.\)
Mathematics Subject Classification ID: 32A30
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 32A10
zbMATH DE Number: 3941913
zbMATH Keywords: holomorphic functions in the unit ball in \({\mathbb{C}}^ n\)
zbMATH Keywords: interpolation by Lipschitz holomorphic functions in the unit sphere in \({\mathbb{C}}^ n\)
reviewed by: Karol Baron
MaRDI profile type: Publication
cites work: Q3969178
cites work: Q3958740
cites work: Q4161569
description: scientific article; zbMATH DE number 3941913