Functions invariant under the Berezin transform
type: Item
instance of: scholarly article
title: Functions invariant under the Berezin transform
zbMATH Open document ID: 0807.47017
published in: Journal of Functional Analysis
publication date: 1994-07-04T00:00:00Z
review text: The author shows that a bounded function \(f\) satisfies \(Bf= f\), where \(B\) is the Berezin transform on the unit disc, if and only if \(f\) is harmonic. There is an equivalent formulation of this result [\textit{S. Axler} and \textit{Ž. Čučković}, Integral Equations Oper. Theory 14, No. 1, 1-12 (1991; Zbl 0733.47027); \textit{W. Rudin}, ``Function theory in the unit ball of \(\mathbb{C}^ N\)'' (1980; Zbl 0495.32001)]: If \(f\) is bounded and satisfies the invariant version of the area mean value property, then \(f\) is harmonic. The main tool employed is Fourier analysis on the Lie group of all Möbius transformations.
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 599764
zbMATH Keywords: Berezin transform on the unit disc
zbMATH Keywords: invariant version of the area mean value property
zbMATH Keywords: Fourier analysis
zbMATH Keywords: Lie group of all Möbius transformations
author: Miroslav Engliš
MaRDI profile type: Publication
full work available at URL: jfan.1994.1048
OpenAlex ID: W2027650067
DOI: 10.1006/JFAN.1994.1048
description: scientific article; zbMATH DE number 599764