The convergence of the Pizzetti series in potential theory
type: Item
instance of: scholarly article
title: The convergence of the Pizzetti series in potential theory
zbMATH Open document ID: 0770.31002
DOI: 10.1016/0022-247X(92)90362-H
author: Ülkü Kuran
author: D. H. Armitage
published in: Journal of Mathematical Analysis and Applications
publication date: 1993-04-01T00:00:00Z
review text: If \(f\) is real-analytic on the unit ball \(B(1)\) of \(\mathbb{R}^ N\) \((N\geq 2)\), then the Pizzetti series of \(f\) is defined by \[ {\mathcal P}(f,r)=\sum^ \infty_{j=0}\{2^ jj!N(N+2)\cdots(N+2j-2)\}^{- 1}\Delta^ jf(O)r^{2j}, \] where \(\Delta^ j\) denotes the iterated Laplacian. This series has a positive radius of convergence. Further, if \(0<r<1\) and \({\mathcal P}(f,r)\) converges, then \({\mathcal P}(f,r)={\mathcal M}(f,r)\), where \({\mathcal M}(f,r)\) denotes the mean of \(f\) over the sphere of centre \(O\) and radius \(r\). This paper gives more precise information on the radius of convergence when \(f\) is a power of a positive harmonic function. It is shown that, if \(h\) is positive and harmonic on \(B(1)\) and \(p\) is a real number, then \({\mathcal P}(h^ p,r)={\mathcal M}(h^ p,r)\) when \(0<r<\tan(\pi/(2N))\). The sharpness of this result is illustrated by several related theorems.
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 30B10
zbMATH DE Number: 150711
zbMATH Keywords: Pizzetti series
zbMATH Keywords: radius of convergence
reviewed by: Stephen J. Gardiner
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 150711