## Some remarks on the Pompeiu problem for groups

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- by David Scott and Alladi Sitaram
- Proc. Amer. Math. Soc.
**104**(1988), 1261-1266 - DOI: https://doi.org/10.1090/S0002-9939-1988-0931747-0
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## Abstract:

A Borel set $E$ in a topological group $G$ is said to be a $P$-set for the space of integrable functions on $G$ if the zero function is the only integrable function whose integral over all left and right translates of $E$ by elements of $G$ is zero. For a "sufficiently nice" group $G$ and a Borel set $E$ of finite Haar measure a certain condition on the Fourier transform of a function related to $E$ is shown to be a sufficient condition for $E$ to be a $P$-set. This condition is then applied to several classes of groups including certain compact groups, certain semisimple Lie groups, the Heisenberg groups and the Euclidean motion group of the plane.## References

- S. C. Bagchi and A. Sitaram,
*Determining sets for measures on $\textbf {R}^{n}$*, Illinois J. Math.**26**(1982), no. 3, 419–422. MR**658452** - Carlos Alberto Berenstein,
*An inverse spectral theorem and its relation to the Pompeiu problem*, J. Analyse Math.**37**(1980), 128–144. MR**583635**, DOI 10.1007/BF02797683 - Carlos A. Berenstein,
*Spectral synthesis on symmetric spaces*, Integral geometry (Brunswick, Maine, 1984) Contemp. Math., vol. 63, Amer. Math. Soc., Providence, RI, 1987, pp. 1–25. MR**876311**, DOI 10.1090/conm/063/876311 - Carlos A. Berenstein and Mehrdad Shahshahani,
*Harmonic analysis and the Pompeiu problem*, Amer. J. Math.**105**(1983), no. 5, 1217–1229. MR**714774**, DOI 10.2307/2374339 - Carlos A. Berenstein and Lawrence Zalcman,
*Pompeiu’s problem on symmetric spaces*, Comment. Math. Helv.**55**(1980), no. 4, 593–621. MR**604716**, DOI 10.1007/BF02566709 - Leon Brown, Bertram M. Schreiber, and B. Alan Taylor,
*Spectral synthesis and the Pompeiu problem*, Ann. Inst. Fourier (Grenoble)**23**(1973), no. 3, 125–154 (English, with French summary). MR**352492** - Ronald L. Lipsman,
*Group representations*, Lecture Notes in Mathematics, Vol. 388, Springer-Verlag, Berlin-New York, 1974. A survey of some current topics. MR**0372116** - Inder K. Rana,
*Determination of probability measures through group actions*, Z. Wahrsch. Verw. Gebiete**53**(1980), no. 2, 197–206. MR**580913**, DOI 10.1007/BF01013316 - Inder K. Rana,
*Determination of probability measures through group actions*, Z. Wahrsch. Verw. Gebiete**53**(1980), no. 2, 197–206. MR**580913**, DOI 10.1007/BF01013316 - N. A. Sapogov,
*A uniqueness problem for finite measures in Euclidean spaces*, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)**41**(1974), 3–13, 165 (Russian). Problems in the theory of probability distributions, II. MR**0385946** - Alladi Sitaram,
*Fourier analysis and determining sets for Radon measures on $\textbf {R}^{n}$*, Illinois J. Math.**28**(1984), no. 2, 339–347. MR**740622** - Alladi Sitaram,
*Some remarks on measures on noncompact semisimple Lie groups*, Pacific J. Math.**110**(1984), no. 2, 429–434. MR**726500** - Mitsuo Sugiura,
*Unitary representations and harmonic analysis*, Kodansha, Ltd., Tokyo; Halsted Press [John Wiley & Sons, Inc.], New York-London-Sydney, 1975. An introduction. MR**0498995** - Michael E. Taylor,
*Noncommutative harmonic analysis*, Mathematical Surveys and Monographs, vol. 22, American Mathematical Society, Providence, RI, 1986. MR**852988**, DOI 10.1090/surv/022
G. Warner, - Yitzhak Weit,
*On Schwartz’s theorem for the motion group*, Ann. Inst. Fourier (Grenoble)**30**(1980), no. 1, vi, 91–107 (English, with French summary). MR**576074** - Lawrence Zalcman,
*Offbeat integral geometry*, Amer. Math. Monthly**87**(1980), no. 3, 161–175. MR**562919**, DOI 10.2307/2321600

*Harmonic analysis on semisimple Lie groups*, 2 vols., Springer-Verlag, 1972.

## Bibliographic Information

- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**104**(1988), 1261-1266 - MSC: Primary 43A60; Secondary 22E30, 43A80
- DOI: https://doi.org/10.1090/S0002-9939-1988-0931747-0
- MathSciNet review: 931747