A Blaschke-type product and random zero sets for Bergman spaces
type: Item
instance of: scholarly article
title: A Blaschke-type product and random zero sets for Bergman spaces
zbMATH Open document ID: 0764.30029
DOI: 10.1007/BF02384861
author: Gregory Bomash
published in: Arkiv för Matematik
publication date: 1993-01-16T00:00:00Z
review text: This paper provides significant new information about zero sets of analytic functions which are in the Bergman spaces \(\mathbb{A}^ p\) of functions in the unit disc. For each \(\lambda\), \(|\lambda|<1\) a Blaschke-type factor \(f_ \lambda\) is introduced which vanishes at \(\lambda\) and is a contractive zero divisor in \(\mathbb{A}^ 2\): \(\| g/f\|_ 2\leq\| g\|_ 2\) whenever \(g\in\mathbb{A}^ 2\), \(g(\lambda)=0\). Further, if \(f\not\equiv 0\in\mathbb{A}^ 2\), then the infinite product \(B\), of the \(f_ \lambda\), as \(\lambda\) ranges over the zeros of \(f\), converges. Thus we have \(f=BF\), with \(F\neq 0\), \(\| F\|_ 1\leq\| f\|_ 2\). The main result, Theorem 2.11, asserts that if a discrete sequence \(\Lambda=\{\lambda_ n\}\) grows sufficiently slowly, then for almost all independent choices of \(\{\theta_ n\}\), the sequence \(\{\lambda_ n e^{i\theta_ n}\}\) is an \(\mathbb{A}^ p\)-zero set, if \(1\leq p\leq 2\) (for \(p>2\) slightly less precise conditions are given). The author poses interesting conjectures in \S3 concerning the best possible growth allowed of such \(\Lambda\), and a refinement of a zero-one law. The paper contrasts this work with earlier work of \textit{C.Horowitz} [Duke Math. J. 41, 693-710 (1974; Zbl 0293.30035)] and \textit{H. Hedenmalm} [J. Reine Angew. Math. 422, 45-68 (1991; Zbl 0734.30040)]. The Blaschke-type product introduced here differ from those of Hedenmalm that the latter considers an extremal problem over a full sequence \(\Lambda\), rather than as a product of individual factors.
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 91590
zbMATH Keywords: random series
zbMATH Keywords: Blaschke product
zbMATH Keywords: Bergman spaces
reviewed by: David Drasin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 91590