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Non-Euclidean Analysis

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Non-Euclidean Geometries

Part of the book series: Mathematics and Its Applications ((MAIA,volume 581))

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Bibliography

  1. J. Anker, The spherical Fourier transform of rapidly decreasing functions-a simple proof of a characterization due to Harish-Chandra, Helgason, Trombi and Varadarajan. J. Funct. Anal. 96 (1991), 331–349.

    Article  MATH  MathSciNet  Google Scholar 

  2. S.C. Bagchi and A. Sitaram, Spherical mean-periodic functions on semisimple Lie groups. Pac. J. Math. 84 (1979), 241–250.

    MathSciNet  MATH  Google Scholar 

  3. Ban, van den, E.P. and H. Schlichtkrull, Asymptotic expansions and boundary values of eigenfunctions on a Riemannian symmetric space. J. Reine Angew. Math. 380 (1987), 108–165.

    MathSciNet  MATH  Google Scholar 

  4. W.H. Barker, L p harmonic analysis on SL(2, R). Memoir of Amer. Math. Soc. 393 Providence, R.I. 1988.

    Google Scholar 

  5. E. Beltrami, Saggio di interpretazione della geometria non euclidea. Giornale di Matematica, 1868.

    Google Scholar 

  6. C.A. Berenstein and B. Rubin, Radon transform of L p-functions on the Lobachevsky space and hyperbolic wavelet transforms. Forum. Math. 11 (1999), 567–590.

    Article  MathSciNet  MATH  Google Scholar 

  7. C.A. Berenstein and B. Rubin, Totally geodesic Radon transform of L p-functions on real hyperbolic space. (Preprint).

    Google Scholar 

  8. C.A. Berenstein and E. Casadio Tarabusi, Inversion formulas for the k-dimensional Radon transform in real hyperbolic spaces. Duke Math. J. 62 (1991), 613–631.

    Article  MathSciNet  MATH  Google Scholar 

  9. C.A. Berenstein, A. Kurusa and E. Casadio Tarabusi, Radon transform on spaces of constant curvature. Proc. Amer. Math. Soc. 125 (1997), 455–461.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Bolyai, The Science Absolute of Space, 1831.

    Google Scholar 

  11. M. Cowling and J. Price, Generalizations of Heisenberg’s inequality. Lecture Notes No. 992, Springer-Verlag, 1983.

    Google Scholar 

  12. M. Cowling, A. Sitaram and M. Sundari, Hardy’s uncertainty principle on semisimple Lie groups. Pac. J. Math. 192 (2000), 293–296.

    MathSciNet  MATH  Google Scholar 

  13. M. Eguchi, Asymptotic Expansions of Eisenstein Integrals and Fourier Transform on Symmetric Spaces. J. Funct. Anal. 34 (1979), 167–216.

    Article  MATH  MathSciNet  Google Scholar 

  14. L. Ehrenpreis and F. Mautner, Some properties of the Fourier transform on semisimple Lie groups I. Ann. of Math. 61 (1955), 406–439.

    Article  MathSciNet  Google Scholar 

  15. B. Fuglede, An integral formula. Math. Scand. 6 (1958), 207–212.

    MATH  MathSciNet  Google Scholar 

  16. I.M. Gelfand and G.E. Shilov, Generalized Functions, I, Academic Press, New York, 1964.

    Google Scholar 

  17. Harish-Chandra, Spherical functions on a semisimple Lie group II. Amer. J. Math. 80 (1958), 553–613.

    MathSciNet  Google Scholar 

  18. G.H. Hardy, A theorem concerning Fourier transforms. J. London Math. Soc. 8 (1933), 227–231.

    MATH  Google Scholar 

  19. S. Helgason, Differential operators on homogeneous spaces. Acta Math. 102 (1959), 239–299.

    MATH  MathSciNet  Google Scholar 

  20. S. Helgason, The Radon transform on Euclidean spaces, two-point homogeneous spaces and Grassmann manifolds. Acta Math. 113 (1965), 153–180.

    Article  MATH  MathSciNet  Google Scholar 

  21. S. Helgason, Radon-Fourier transforms on symmetric spaces and related group representations. Bull. Amer. Math. Soc, 71 (1965), 757–763.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. S. Helgason, The surjectivity of invariant differential operators on symmetric spaces. Ann. of Math. 98 (1973), 451–480.

    Article  MATH  MathSciNet  Google Scholar 

  24. S. Helgason, Eigenspaces of the Laplacian; integral representations and irreducibility. J. Functional Anal. 17 (1974), 328–353.

    Article  MATH  MathSciNet  Google Scholar 

  25. S. Helgason, A duality for symmetric spaces with applications to group representations II. Differential equations and eigenspace representations. Advan. Math. 22 (1976), 187–219.

    Article  MATH  MathSciNet  Google Scholar 

  26. S. Helgason, Groups and Geometric Analysis. Acad. Press, 1984, Amer. Math. Soc., 2000.

    Google Scholar 

  27. S. Helgason, Geometric Analysis on Symmetric Spaces. Math Surveys and Monographs No. 39, AMS, Providence, R.I. 1994.

    MATH  Google Scholar 

  28. S. Helgason, The Radon Transform. Birkhäuser, Boston, 1999.

    MATH  Google Scholar 

  29. S. Helgason, R. Rawat, J. Sengupta and A. Sitaram, Some remarks on the Fourier transform on a symmetric space. Tech. Report, Ind. Stat. Inst. Bangalore 1998.

    Google Scholar 

  30. S. Helgason and J. Sengupta, Preprint 1997.

    Google Scholar 

  31. S. Ishikawa, The range characterization of the totally geodesic Radon transform on the real hyperbolic space. Duke Math. J. 90 (1997), 149–203.

    Article  MATH  MathSciNet  Google Scholar 

  32. F. John, Plane Waves and Spherical Means. Wiley, New York, 1955.

    MATH  Google Scholar 

  33. S.R. Jensen, Sufficient conditions for the inversion formula for the k-plane transform in R n. (Preprint).

    Google Scholar 

  34. A. Kurusa, The Radon transform on hyperbolic space. Geom. Dedicata 40 (1991), 325–339.

    Article  MATH  MathSciNet  Google Scholar 

  35. J. Lewis, Eigenfunctions on symmetric spaces with distribution-valued boundary forms. J. Funct. Anal. 29 (1978), 287–307.

    Article  MATH  MathSciNet  Google Scholar 

  36. S. Lissianoi and I. Ponomarev, On the inversion of the geodesic Radon transform on the hyperbolic plane. Inverse Problems 13 (1997), 1053–1062.

    Article  MathSciNet  MATH  Google Scholar 

  37. N. Lobatchevski, Geometrical Researches on the Theory of Parallels, Kasan, 1826.

    Google Scholar 

  38. G.W. Morgan, A note on Fourier transforms. J. London Math. Soc. 9 (1934), 187–192.

    MATH  Google Scholar 

  39. P. Mohanty, S.K. Ray, R.P. Sarkar and A. Sitaram, Helgason Fourier Transform for Symmetric Spaces II, J. Lie Theory 14 (2004), 227–242.

    MathSciNet  MATH  Google Scholar 

  40. E.K. Narayanan and S.K. Ray, L p version of Hardy’s theorem on semisimple Lie groups. Proc. Amer. Math. Soc. 130 (2002), 1859–1866.

    Article  MathSciNet  MATH  Google Scholar 

  41. T. Oshima, Y. Saburi and M. Wakayama, Paley-Wiener theorems on a symmetric space and their application. Diff. Geom. and Appl. (1991), 247–278.

    Google Scholar 

  42. J. Radon, Über die Bestimmung von Finktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber. Verth. Sächs. Akad. Wiss. Leipzig. Math. Nat. kl. 69 (1917), 262–277.

    MATH  Google Scholar 

  43. S.K. Ray and R.P. Sarkar, Cowling-Price theorem and characterization of heat kernel of symmetric spaces. (Preprint).

    Google Scholar 

  44. F. Rouvière, Inverting Radon transforms; the group-theoretic approach. Enseign. Math. 47 (2001), 205–252.

    MATH  MathSciNet  Google Scholar 

  45. B. Rubin, Reconstruction of functions from their integrals over k-planes. (Preprint).

    Google Scholar 

  46. B. Rubin. Helnason-Marchand inversion formulas for Radon transforms. Proc. Amer. Math. Soc. 130 (2002), 3017–3023.

    Article  MATH  MathSciNet  Google Scholar 

  47. B. Rubin, Radon, Cosine and Sine transforms on real hyperbolic spaces. Advan. Math. 170 (2002) 206–233.

    MATH  Google Scholar 

  48. R. Sarkar and A. Sitaram, The Helgason Fourier Transform for Symmetric Spaces. A tribute to C.S. Seshadri. A collection of articles on geometry and representation theory. Basel: Birkhuser. Trends in Mathematics, (2003), 467–473.

    Google Scholar 

  49. R.P. Sarkar, Wiener Tauberian theorem for rank one symmetric spaces. Pacific J. of Math. 186 (1998), 349–358.

    Article  MATH  MathSciNet  Google Scholar 

  50. L. Schwartz, Théorie generale des fonctions moyenne-periodiques. Ann, of Math. 48 (1947), 857–929.

    Article  MATH  MathSciNet  Google Scholar 

  51. J. Sengupta, The Uncertainty Principle on Riemannian symmetric spaces of the noncompact type. Proc. Amer. Math. Soc. 128 (2000) 2493–2499.

    Article  MATH  MathSciNet  Google Scholar 

  52. V.I. Semyanisty, Homogeneous functions and some problems of integral geometry in spaces of constant curvature. Soviet Math. Dokl. 2 (1961) 59–62.

    MathSciNet  Google Scholar 

  53. N. Shimeno, An analog of Hardy’s theorem for the Harish-Chandra transform. Hiroshima Math. J. 31 (2001), 383–390.

    MATH  MathSciNet  Google Scholar 

  54. A. Sitaram, An analog of the Wiener Tauberian theorem for spherical transforms on semisimple Lie groups. Pac. J. Math. 89 (1980) 439–445.

    MATH  MathSciNet  Google Scholar 

  55. A. Sitaram, On an analog of the Wiener Tauberian theorem for symmetric spaces of the noncompact type. Pac. J. Math. 133 (1988) 197–208.

    MATH  MathSciNet  Google Scholar 

  56. A. Sitaram and M. Sundari, An analog of Hardy’s theorem for very rapidly decreasing functions on semisimple Lie groups. Pac. J. Math. 177 (1997), 187–200.

    MathSciNet  MATH  Google Scholar 

  57. R.J. Stanton and P.A. Thomas, Pointwise inversion of the spherical transform on L p(G/K) (1 ≤ p < 2). Proc. Amer. Math. Soc. 73 (1979), 398–404.

    Article  MathSciNet  MATH  Google Scholar 

  58. P. Trombi and V.S. Varadarajan, Spherical transforms on a semisimple Lie group. Ann. of Math. 94 (1971), 246–303.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139

    Sigurdur Helgason

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  1. Sigurdur Helgason
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Editors and Affiliations

  1. Rutgers Center for Operations Research, Piscataway, New Jersey, USA

    András Prékopa

  2. Budapest University of Technology and Economics, Hungary

    Emil Molnár

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Helgason, S. (2006). Non-Euclidean Analysis. In: Prékopa, A., Molnár, E. (eds) Non-Euclidean Geometries. Mathematics and Its Applications, vol 581. Springer, Boston, MA. https://doi.org/10.1007/0-387-29555-0_18

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