Zero sets and uniqueness sets with one cluster point for the Dirichlet space
type: Item
instance of: scholarly article
title: Zero sets and uniqueness sets with one cluster point for the Dirichlet space
zbMATH Open document ID: 1189.31013
author: Javad Mashreghi
author: M. Shabankhah
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-07-09T00:00:00Z
review text: Let \((z_n)\) be a sequence of points on the open unit disk \(\mathbb{D}\). In a classical paper, \textit{H. Shapiro} and \textit{A. Shields} [Math. Z. 80, 217--229 (1962; Zbl 0115.06301)] have shown that if \(\sum(-\log(1-|z_n|))^{-1}\) is finite, then \((z_n)\) is a zero sequence for the Dirichlet space, thus improving a previous result of L. Carleson. The present authors prove the following (Theorem 2.4). Let \(\varphi:[0,1]\to[0,\infty)\) be an increasing function such that \(\int_0^1\varphi(t)t^{-2}dt<\infty\), and suppose that there exists a non-zero function \(f\) in the Dirichlet space such that \(|f(\zeta)|^2\leq\varphi(|1-\zeta|/2)\) for almost every \(\zeta\) on the unit circle. Let \((z_n)\) be a Blaschke sequence on \(\mathbb{D}\) such that \(\sum\varphi(|1-z_n|)\) is finite. Then \((z_n)\) is a zero sequence for the Dirichlet space. The proof uses the theorem by Shapiro and Shields mentioned above and some preliminary steps. The authors use their results to construct families of zero sets for the Dirichlet space by giving suitable functions \(\varphi\). One of the examples is \(\varphi(t)=\exp(-\beta t^{-\alpha})\) where \(0<\alpha<1\) and \(\beta>0\).
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 30J10
zbMATH DE Number: 5577087
zbMATH Keywords: zero sets
zbMATH Keywords: uniqueness sets
zbMATH Keywords: Dirichlet space
zbMATH Keywords: Blaschke products
reviewed by: António Serra
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.04.023
OpenAlex ID: W1982679139
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DOI: 10.1016/J.JMAA.2009.04.023
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description: scientific article; zbMATH DE number 5577087