The Pompeiu problem and discrete groups

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• 作者：Michael J. Puls (1)
  
• 关键词：Abelian group ; Free group ; Harmonic function ; Left translates ; Mean ; value property ; Pompeiu property ; Radial functions ; Torsion free rank ; Zero divisors ; Primary 43A15 ; Secondary 43A32 ; 43A70 ; 43A80 ; 43A90
• 刊名：Monatshefte f¨1r Mathematik
• 出版年：2013
• 出版时间：December 2013
• 年：2013
• 卷：172
• 期：3-4
• 页码：415-429
• 全文大小：
• 作者单位：Michael J. Puls (1)
  
  1. Department of Mathematics, John Jay College-CUNY, 524 West 59th Street, New York, NY, 10019, USA
  
• ISSN：1436-5081

文摘

We formulate a version of the Pompeiu problem in the discrete group setting. Necessary and sufficient conditions are given for a finite collection of finite subsets of a discrete abelian group, whose torsion free rank is less than the cardinal of the continuum, to have the Pompeiu property. We also prove a similar result for nonabelian free groups. A sufficient condition is given that guarantees the harmonicity of a function on a nonabelian free group if it satisfies the mean-value property over two spheres.