Some remarks on the Pompeiu problem for groups

Scott, David; Sitaram, Alladi

doi:10.1090/S0002-9939-1988-0931747-0

Some remarks on the Pompeiu problem for groups
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by David Scott and Alladi Sitaram PDF

Proc. Amer. Math. Soc. 104 (1988), 1261-1266 Request permission

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

Harmonic analysis on semisimple Lie groups

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