Successive averages and harmonic functions
type: Item
instance of: scholarly article
title: Successive averages and harmonic functions
zbMATH Open document ID: 0879.31002
DOI: 10.1007/BF02788028
author: Ivan Netuka
author: Wolfhard Hansen
published in: Journal d'Analyse Mathématique
publication date: 1998-01-11T00:00:00Z
review text: For every \(r>0\) let \(m_r\) be the normalized Lebesgue measure on the ball \(B(0, r)\) in \(\mathbb{R}^d\). This paper is motivated by the following question asked by G. Choquet in 1994: Let \(f\) be a continuous real function on \(\mathbb{R}^d\) and let \(r_1\), \(r_2\), \(r_3\), \(\dots\) be strictly positive real numbers; under what conditions on \(f\) and the sequence \((r_n)\) does \((f\ast m_{r_1}\ast m_{r_2}\ast\cdots\ast m_{r_n})\) converge to a harmonic function? Among other things, the authors prove the following result. If \(\sum^{\infty}_{n=1}r^2_n=\infty\), then the following holds: (1) for every superharmonic function \(u\) on \(\mathbb{R}^d\) admitting a harmonic minorant the sequence \((u\ast m_{r_1}\ast m_{r_2} \ast\cdots\ast m_{r_n})\) decreases locally uniformly to the greatest harmonic minorant of \(u\); (2) for every continuous \(f\) with comapct support the sequence \((f\ast m_{r_1}\ast m_{r_2}\ast\cdots\ast m_{r_n})\) converges uniformly to zero; and (3) for every bounded \(f\) such that the average \(\int fdm_r\) tends to some real number \(A\) as \(r\rightarrow\infty\), the sequence \((f\ast m_{r_1}\ast m_{r_2}\ast\cdots\ast m_{r_n})\) converges locally uniformly to \(A\). The paper also contains similar results when \(m_r\) is replaced by more general measures.
Mathematics Subject Classification ID: 31B05
zbMATH DE Number: 1038444
zbMATH Keywords: harmonic functions
zbMATH Keywords: superharmonic functions
zbMATH Keywords: harmonic minorant
zbMATH Keywords: convolution
reviewed by: Renming Song
MaRDI profile type: Publication
cites work: Probability theory. Translated from the German by Robert B. Burckel
cites work: Functions satisfying the mean value property in the limit
cites work: Potential theory. An analytic and probabilistic approach to balayage
description: scientific article; zbMATH DE number 1038444