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The Pompeiu Problem on locally symmetric spaces

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References

  1. E. Badertscher,Harmonic analysis and Radon transforms on pencils of geodesics, Math. Scand.58 (1986), 187–214.

    MATH  MathSciNet  Google Scholar 

  2. S. C. Bagchi and A. Sitaram,The Pompeiu problem revisited, L’Enseignement Math.36 (1990), 67–91.

    MATH  MathSciNet  Google Scholar 

  3. C. A. Berenstein and M. Shahshahani,Harmonic analysis and the Pompeiu problem, Am. J. Math.105 (1983), 1217–1229.

    Article  MATH  MathSciNet  Google Scholar 

  4. C. A. Berenstein and L. Zalcman,Pompeiu’s problem on spaces with constant curvature, J. Analyse Math.30 (1976), 113–130.

    Article  MATH  MathSciNet  Google Scholar 

  5. C. A. Berenstein and L. Zalcman,Pompeiu’s problem on symmetric spaces, Comment. Math. Helv.55 (1980), 593–621.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. L. Besse,Manifolds all of whose Geodesics are Closed, Springer-Verlag, Berlin, 1978.

    MATH  Google Scholar 

  7. R. P. Boas,Entire Functions, Academic Press, New York, 1954.

    MATH  Google Scholar 

  8. J. R. Brown,Ergodic Theory and Toplogical Dynamics, Academic Press, New York, 1976.

    Google Scholar 

  9. L. Brown, B. M. Schreiber and B. A. Taylor,Spectral synthesis and the Pompeiu problem, Ann. Inst. Fourier (Grenoble)23(3) (1973), 125–154.

    MathSciNet  Google Scholar 

  10. L. Chakalov,Sur un problème de D. Pompeiu, Annaire Univ. Sofia Fac. Phys. Math.40(Livre 1) (1944), 1–44.

    Google Scholar 

  11. I. Chavel,Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984.

    MATH  Google Scholar 

  12. P. Funk,Über Flächen mit Tauter geschlossenen geodätischen Linien, Math. Annalen74 (1913), 278–300.

    Article  MATH  MathSciNet  Google Scholar 

  13. P. Funk,Über eine geometrische Anwendung der Abelschen Integralgleichung, Math. Annalen77 (1916), 129–135.

    Article  MathSciNet  Google Scholar 

  14. R. Gangolli and V. S. Varadarajan,Harmonic Analysis of Spherical Functions on Real Reductive Groups, Springer Verlag, Berlin, 1988.

    MATH  Google Scholar 

  15. P. Günther,Sphärische Mittelwerte in kompakten harmonischen Riemannschen Mannigfaltigkeiten, Math. Annalen165 (1966), 281–296.

    Article  MATH  Google Scholar 

  16. S. Helgason,Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978.

    MATH  Google Scholar 

  17. S. Helgason,Groups and Geometric Analysis, Academic Press, New York, 1984.

    MATH  Google Scholar 

  18. R. Hochreuter,Integraltransformationen vom Radonschen Typ auf der Sphäre, Dissertation, Universität Erlangen, 1989.

  19. B. Hoogenboom,Intertwining Functions on Compact Lie Groups, Dissertation, University of Leiden, 1983.

  20. F. John,Abhängigkeit zwischen den Flächenintegralen einer stetigen Funktion, Math. Annalen111 (1938), 541–559.

    Article  Google Scholar 

  21. S. Kobayashi,Transformation Groups in Differential Geometry, Springer Verlag, Berlin, 1972.

    MATH  Google Scholar 

  22. T. H. Koornwinder,Jacobi functions and analysis on noncompact semisimple Lie groups, inSpecial Functions, Group Theoretical Aspects and Applications, R. A. Askey, T. H. Koornwinder and W. Schempp (eds.), Reidel, Dordrecht, 1984, pp. 1–85.

    Google Scholar 

  23. P. D. Lax and R. S. Phillips,The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, J. Funct. Anal.46 (1982), 280–350.

    Article  MATH  MathSciNet  Google Scholar 

  24. W. Müller,Manifolds with Cusps of Rank One, Lecture Notes in Math.1244, Springer-Verlag, Berlin, 1987.

    MATH  Google Scholar 

  25. V. Pati, M. Shahshahani and A. Sitaram,On the spherical mean value operator for compact symmetric spaces, Technical Report, Indian Statistical Institute, 1989.

  26. D. Pompeiu,Sur une propriété des functions continues dépendant de plusieurs variables, Bull. Sci. Math.53(2) (1929), 328–332.

    Google Scholar 

  27. J. Radon,Über die Bestimmung von Funktionen durch ihre Integralwerte längs bestimmter Mannigfaltigkeiten, Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math. Nat. Kl.69 (1917), 262–277.

    Google Scholar 

  28. I. K. Rana,Determination of probability measures through group actions, Z. Wahrsch. verw. Gebiete53 (1980), 197–206.

    Article  MATH  MathSciNet  Google Scholar 

  29. R. Schneider,Functions on a sphere with vanishing integrals over certain subspheres, J. Math. Anal. Appl.26 (1969), 381–384.

    Article  MATH  MathSciNet  Google Scholar 

  30. R. Schneider,Über eine Integralgleichung in der Theorie der konvexen Körper, Math. Nachr.44 (1970), 55–75.

    Article  MATH  MathSciNet  Google Scholar 

  31. A. Sitaram,Some remarks on measures on noncompact semisimple Lie groups, Pacific J. Math.110 (1984), 429–434.

    MATH  MathSciNet  Google Scholar 

  32. T. Sunada,Spherical means and geodesic chains on a Riemannian manifold, Trans. Am. Math. Soc.267 (1981), 483–501.

    Article  MATH  MathSciNet  Google Scholar 

  33. T. Sunada,Geodesic flows and geodesic random walks, inGeometry of Geodesics and Related Topics, Volume 3 ofAdvanced Studies in Pure Mathematics, K. Shiohama (ed.), North-Holland, Amsterdam, 1984, pp. 47–85.

    Google Scholar 

  34. P. Ungar,Freak theorem about functions on a sphere, J. London Math. Soc.29 (1954), 100–103.

    Article  MATH  MathSciNet  Google Scholar 

  35. L. Vretare,Elementary spherical functions on symmetric spaces, Math. Scand.39 (1976), 343–358.

    MathSciNet  Google Scholar 

  36. S. A. Williams,A partial solution of the Pompeiu problem, Math. Annalen223 (1976), 183–190.

    Article  MATH  Google Scholar 

  37. S. A. Williams,Analyticity of the boundary for Lipschitz domains without the Pompeiu property, Indiana Univ. Math. J.,30 (1981), 357–369.

    Article  MATH  MathSciNet  Google Scholar 

  38. J. A. Wolf,Spaces of Constant Curvature, McGraw-Hill, New York, 1967.

    MATH  Google Scholar 

  39. L. Zalcman,Analyticity and the Pompeiu problem, Arch. Rational Mech. Anal.47 (1972), 237–254.

    Article  MATH  MathSciNet  Google Scholar 

  40. L. Zalcman,Offbeat integral geometry, Am. Math. Monthly87 (1980), 161–175.

    Article  MATH  MathSciNet  Google Scholar 

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Badertscher, E. The Pompeiu Problem on locally symmetric spaces. J. Anal. Math. 57, 250–281 (1991). https://doi.org/10.1007/BF03041072

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