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Inversion de la transformation de Pompeiu locale dans les espaces hyperboliques reel et complexe: Cas des deux boules

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Abstract

In this paper we extend the result we have established for the hyperbolic disk in [8] to the real and complex hyperbolic spaces. This includes the reconstruction of a function defined in a fixed ball from its averages on balls of radiir 1,r 2 lying inB(0,R).

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References

  1. R. Askey and J. Fitch,Integral representations for Jacobi polynomials and some applications, J. Math. Anal. Appl.26 (1969), 411–437.

    Article  MATH  MathSciNet  Google Scholar 

  2. C. A. Berenstein and R. Gay,A local version of the two circles theorem, Isr. J. Math.55 (1986), 267–288.

    Article  MATH  MathSciNet  Google Scholar 

  3. C. A. Berenstein, R. Gay and A. Yger,The three squares theorem, a local version, Festschrift for M. Cotlar, Lecture Notes in Applied Mathematics122 (1990), 253–259.

    MathSciNet  Google Scholar 

  4. C. A. Berenstein R. Gay and A. Yger,Inversion, of the local Pompeiu transform, J. Analyse Math.,55 (1980), 259–287.

    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. L. Brown, B. M. Schreiber and B. A. Taylor,Spectral synthesis and Pompeiu problem, Ann. Inst. Fourier23 (1973), 125–154.

    MathSciNet  Google Scholar 

  7. M. Eguchi, M. Hashizume and K. Okamoto,The Paley-Wiener theorem for distributions on symmetric spaces, Hiroshima Math. J.3 (1973), 109–120.

    MATH  MathSciNet  Google Scholar 

  8. M. El Harchaoui,Inversion de la transformation de Pompéiu dans le disque hyperbolique, Publicacions Matemàtiques37 (1993), 133–164.

    MATH  Google Scholar 

  9. A. Erdélyi (ed.),Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York, 1953.

    Google Scholar 

  10. M. Flensted-Jensen,Paley-Wiener type theorems for a differential operator connected with symmetric spaces, Arkiv Math.10 (1972), 143–164.

    Article  MATH  MathSciNet  Google Scholar 

  11. M. Flensted-Jensen and T. H. Koornwinder,Positive definite functions on a non-compact, rank one symmetric space, Lecture Notes in Math.,239 (1979), 249–282.

    Article  MathSciNet  Google Scholar 

  12. S. Helgason,Differential geometry and symmetric spaces, Academic Press, New York, 1962.

    MATH  Google Scholar 

  13. S. Helgason,A duality for symmetric spaces, with applications to group representations, Advances in Math.,5 (1970), 1–154.

    Article  MATH  MathSciNet  Google Scholar 

  14. S. Helgason,Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators and Spherical Functions, Academic Press, New York, 1984.

    MATH  Google Scholar 

  15. K. Johnson and N. Wallach,Composition series and intertwining operators for the spherical series, I. Trans. Amer. Math. Soc.229 (1977), 137–173.

    Article  MATH  MathSciNet  Google Scholar 

  16. T. H. Koornwinder,The addition formula for Jacobi Polynomials II, III. The Laplace type integral representation and the product formula, T. W. Rapport133, 135, Math. Centrum, Amsterdam, 1972.

    Google Scholar 

  17. S. Krantz,Function Theory of Several Complex Variables, Wiley, New York, 1982.

    MATH  Google Scholar 

  18. J. L. Liones et E. Magegns,Problèmes aux limites non homogènes et applications, Vol. 1, Dunod, Paris, 1968.

    Google Scholar 

  19. E. D. Quinto,Pompeiu transform on geodesic spheres in real analytic manifolds, Preprint (1993).

  20. W. Rudin,Function theory in the unit ball of ℂn, Grundlehren der Mathematischen Wissenschaften241, 1980.

  21. R. T. Selley,Spherical harmonics, Amer. Math. Monthly,73 (1966), 115–121.

    Article  Google Scholar 

  22. G. Szegö,Orthogonal Polynomials, Amer. Math. Soc., Providence, R. I., 1968.

    Google Scholar 

  23. G. N. Watson,Asymptotic expansions of hypergeometric functions, Trans. Cambridge, Philos. Soc.22 (1918), 277–308.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. L. Zalcman,A bibliographical survey of the Pompeiu problem, inApproximations by Solutions of Partial Differential Equations (M. Goldstein and W. Haussman, eds.), Kluwer Academic Publishers, Dordrecht, 1992, pp. 185–194.

    Google Scholar 

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Authors and Affiliations

  1. Université Bordeaux I UER Mathématiques, 351, Cours de la Libération, 33405, Talence, France

    Mimoun El Harchaoui

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  1. Mimoun El Harchaoui
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Harchaoui, M.E. Inversion de la transformation de Pompeiu locale dans les espaces hyperboliques reel et complexe: Cas des deux boules. J. Anal. Math. 67, 1–37 (1995). https://doi.org/10.1007/BF02787785

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  • DOI: https://doi.org/10.1007/BF02787785

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