Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berenstein, C.A., Zalcman, L.: Pompeiu’s problem on symmetric spaces. Comment. Math. Helv. 55(4), 593–621 (1980)
Carey, A.L.: Eberhard Kaniuth, and William Moran. The Pompeiu problem for groups. Math. Proc. Cambridge Philos. Soc. 109(1), 45–58 (1991)
Cohen, J.M.: von Neumann dimension and the homology of covering spaces. Quart. J. Math. Oxford Ser. (2) 30(118), 133–142 (1979)
Cohen, J.M., Picardello, M.A.: The 2-circle and 2-disk problems on trees. Israel J. Math. 64(1), 73–86 (1988)
Figà-Talamanca, A., Picardello, M.A.: Harmonic Analysis on Free Groups, Vol. 87 of Lecture Notes in Pure and Applied Mathematics. Marcel Dekker Inc., New York (1983)
Vieli, F.J.G.: A two-spheres problem on homogeneous trees. Combinatorica 18(2), 185–189 (1998)
Laczkovich, M., Székelyhidi, G.: Harmonic analysis on discrete abelian groups. Proc. Am. Math. Soc. 133(6), 1581–1586 (2005) (electronic)
Linnell, P.A.: Zero divisors and group von Neumann algebras. Pacific J. Math. 149(2), 349–363 (1991)
Linnell, P.A.: Zero divisors and \(L^2(G)\). C. R. Acad. Sci. Paris Sér. I Math. 315(1), 49–53 (1992)
Linnell, P.A., Puls, M.J.: Zero divisors and \(L^p(G)\). II. NY J. Math. 7, 49–58 (2001) (electronic)
Peyerimhoff, N., Samiou, E.: Spherical spectral synthesis and two-radius theorems on Damek-Ricci spaces. Ark. Mat. 48(1), 131–147 (2010)
Puls, M.J.: Zero divisors and \(L^p(G)\). Proc. Am. Math. Soc. 126(3), 721–728 (1998)
Rawat, R., Sitaram, A.: The injectivity of the Pompeiu transform and \(L^p\)-analogues of the Wiener-Tauberian theorem. Israel J. Math. 91(1–3), 307–316 (1995)
Scott, D., Sitaram, A.: Some remarks on the Pompeiu problem for groups. Proc. Am. Math. Soc. 104(4), 1261–1266 (1988)
Zalcman, L.: Offbeat integral geometry. Am. Math. Monthly 87(3), 161–175 (1980)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Constantin.
The research for this paper was partially supported by PSC-CUNY Grant 65364-00 43.
Rights and permissions
About this article
Cite this article
Puls, M.J. The Pompeiu problem and discrete groups. Monatsh Math 172, 415–429 (2013). https://doi.org/10.1007/s00605-013-0524-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-013-0524-z
Keywords
- Abelian group
- Free group
- Harmonic function
- Left translates
- Mean-value property
- Pompeiu property
- Radial functions
- Torsion free rank
- Zero divisors