Zeros of extremal functions in weighted Bergman spaces.
type: Item
instance of: scholarly article
title: Zeros of extremal functions in weighted Bergman spaces.
zbMATH Open document ID: 1051.30012
author: Rachel Weir
published in: Pacific Journal of Mathematics
publication date: 2004-09-16T00:00:00Z
review text: Let \(D\) be the open unit disk in the complex plane and let \(dA\) be area measure on \(D\) normalized so that \(D\) has area 1. For \(p>0\) and \(\alpha> -1\) let \(A^p_\alpha\) denote the weighted Bergman space of \(D\) consisting of analytic functions \(f\) in \(D\) such that \[ \| f\|^p_{p,\alpha}= (\alpha+1)\int_D \bigl| f(x)\bigr|^p \bigl(1-| z|^2\bigr)^\alpha dA(z)<\infty. \] The main result of the paper is the following: if \(0<\alpha\leq 1\) and \(Z= \{z_k\}\) is an \(A^p_\alpha\)-zero set and none of \(z_k\) is 0, then the solution of the extremal problem \[ \sup\bigl\{\text{Re}\,f(0):\| f\|_{p, \alpha} \leq 1,\;f(z_k)=0,\;k=1,2,\dots \bigr\} \] does not have any zero other than \(\{z_k\}\). An application to hypergeometric functions is obtained as a consequence.
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 30D55
Mathematics Subject Classification ID: 33C05
Mathematics Subject Classification ID: 46E50
zbMATH DE Number: 2101047
zbMATH Keywords: Bergman space
zbMATH Keywords: zero set
zbMATH Keywords: extremal problem
zbMATH Keywords: contractive zero divisor
reviewed by: Kehe Zhu
MaRDI profile type: Publication
full work available at URL: pjm.2003.208.187
OpenAlex ID: W2095272047
DOI: 10.2140/PJM.2003.208.187
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description: scientific article; zbMATH DE number 2101047