Some conditions on Bergman space zero sets
type: Item
instance of: scholarly article
title: Some conditions on Bergman space zero sets
zbMATH Open document ID: 0795.30006
DOI: 10.1007/BF02835961
author: Charles Horowitz
published in: Journal d'Analyse Mathématique
publication date: 1994-05-18T00:00:00Z
review text: The author extends work of Korenblum and Hedenmalm on the zero sets of the functions in the Bergman spaces \(A^ p\), with concentration on the individual spaces \(A^ p\) rather than on their union. The space \(A^ p\), \(0< p<\infty\) is the collection of functions \(f\) analytic in the unit disk \(U\) and satisfying \[ \| f\|^ p_ p= \textstyle{{1\over \pi}}\int_ U \int | f(z)|^ p dx dy< \infty. \] The author narrows the gap between conditions known to be necessary for a sequence to be the zero set of a function in \(A^ p\) and conditions known to be sufficient. The principle sufficient conditions restrict the number of sequence elements in (roughly) truncated Stolz angles at the boundary of the unit disk. The functions constructed are infinite canonical products with carefully chosen convergence factors.
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 30D50
zbMATH DE Number: 569442
zbMATH Keywords: Jensen's inequality
zbMATH Keywords: Blaschke product
zbMATH Keywords: Bergman spaces
zbMATH Keywords: zero set
reviewed by: James W. Cannon
MaRDI profile type: Publication
cites work: A Blaschke-type product and random zero sets for Bergman spaces
cites work: Q4037154
cites work: A factorization theorem for square area-integrable analytic functions.
cites work: Zeros of functions in the Bergman spaces
cites work: An extension of the Nevanlinna theory
cites work: Q4713937
cites work: A probabilistic zero set condition for the Bergman space
full work available at URL: https://doi.org/10.1007/bf02835961
OpenAlex ID: W2095568302
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description: scientific article; zbMATH DE number 569442