庞培问题与离散群
The Pompeiu Problem and Discrete Groups
摘要：
我们制定了一个版本的庞培问题的离散组设置。给出了离散阿贝尔群的有限子集的有限集合具有Pompeiu性质的充要条件，该群的无挠的秩小于连续统的基数.对于非交换自由群，我们也证明了一个类似的结果。给出了非交换自由群上的函数在两个球面上满足均值性质时是调和函数的一个充分条件。
英文摘要：
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.
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