On the zeros of functions with finite Dirichlet integral
type: Item
instance of: scholarly article
title: On the zeros of functions with finite Dirichlet integral
zbMATH Open document ID: 0854.30003
published in: Kodai Mathematical Journal
publication date: 1996-08-29T00:00:00Z
review text: Let \({\mathcal D}\) be the Dirichlet space of functions analytic in \(\Delta = \{z : |z |< 1\}\) and having finite Dirichlet integral \[ \int_\Delta |f' (z) |^2 dx dy. \] It is known that the zero-set of any function of \({\mathcal D}\) satisfies the Blaschke condition but the converse is not true. The author calls \(A \subset \Delta\) a Blaschke set for \({\mathcal D}\) if any subsequence of \(A\) satisfying the Blaschke condition is the zero-set of a function of \({\mathcal D}\). The main result is the following Theorem. A set \(A \subset \Delta\) is Blaschke for \({\mathcal D}\) if and only if \[ - \int^\pi_{- \pi} \log \text{dist} (A, e^{it}) dt < \infty. \tag{1} \] Moreover, if (1) holds, then there is an outer function \(F \in {\mathcal D}\) with the property that \(BF \in {\mathcal D}\) for every Blaschke product \(B\) with zeros in \(A\).
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 895926
zbMATH Keywords: Dirichlet space
zbMATH Keywords: zero-set
zbMATH Keywords: Blaschke product
author: Krzysztof Bogdan
reviewed by: I. V. Ostrovskii
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138043542
OpenAlex ID: W1971299678
Wikidata QID: Q110598185
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DOI: 10.2996/KMJ/1138043542
description: scientific article; zbMATH DE number 895926