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Radon transforms and wave equations

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1684)

Keywords

  • Wave Equation
  • Symmetric Space
  • Homogeneous Space
  • Inversion Formula
  • Cartan Subgroup

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References

  • [BO] U. Bunke and M. Olbrich, The wave kernel for the Laplacian on classical locally symmetric spaces of rank one, theta functions, trace formulas and the Selberg zeta function, Ann Glob. Anal. Geom. 12 (1994), 357–405.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [BT] C. Berenstein and C. Casadio Tarabusi, Range of the k-dimensional Randon transform in real hyperbolic spaces, Forum Math. 5 (1993), 603–616.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [CH] R. Courant and Hilbert, Methoden der Mathematischen Physik, vol. II (Springer, eds.), Berlin, 1973.

    MATH  Google Scholar 

  • [Co] E. Cotton, Sur les invariant differentiels ..., Ann. Éc. Norm. Sup. 17 (1900), 211–244.

    MathSciNet  MATH  Google Scholar 

  • [CV] O.A. Chalykh and A.P. Veselov, Integrability and Huygens’ principle on symmetric spaces, Preprint (1995).

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • [GG] I.M. Gelfand and M.I. Graev, The geometry of homogeneous spaces, group representations in homogeneous spaces and questions in integral geometry related to them, Amer. Math. Soc. Transl. 37 (1964).

    Google Scholar 

  • [Gi] S.G. Gindikin, Integral geometry on quadrics, Amer. Math. Soc. Transl. (2) 169 (1995), 23–31.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [GK] S.G. Gindikin and F.I. Karpelevic, Plancherel measure of Riemannian symmetric spaces of non-positive curvature, Dokl. Akad. Nauk USSR 145 (1962), 252–255.

    MathSciNet  Google Scholar 

  • [Go] F. Gonzalez, Radon transforms on Grassmann manifolds, J. Funct. Anal. 71 (1987), 339–362.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [GQ] F. Gonzalez and E.T. Quinto, Support theorems for Randon transforms on higher rank symetric spaces, Proc. Amer. Math. Soc. 122 (1994), 1045–1052.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [HC] Harish-Chandra, Spherical functions on a semisimple Lie group I, Amer. J. Math. 80 (1958), 241–310.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [H1] S. Helgason, Differential operators on homogeneous spaces, Acta Math. 102 (1959), 239–299.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [H2]-, A duality in integral geometry; some generalizations of the Radon transform, Bull. Amer. Math. Soc. 70 (1964), 435–446.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [H3]-, Fundamental solutions of invariant differential operators on symmetric spaces Amer. J. Math. 86 (1964), 565–601.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [H4]-, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds, Acta Math. 113 (1965), 153–180.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [H5]-, A duality for symmetric spaces with applications to group representations, Advan. Math. 5 (1970), 1–154.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [H6]-, The surjectivity of invariant differential operators on symmetric spaces, Ann. of Math. 98 (1973), 451–480.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [H7]-, Support of Radon transforms, Advan. Math. 38 (1980), 91–100.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [H8]-, The Radon Transform, Birkhäuser, Boston, 1980.

    CrossRef  MATH  Google Scholar 

  • [H9]-, Groups and Geometric Analysis; Integral Geometry, Invariant Differential Operators and Spherical Functions, Academic Press, New York, 1984.

    MATH  Google Scholar 

  • [H10]-, The totally geodesic transform on constant curvature spaces, Amer. Math. Soc. 113 (1990), 141–149.

    MathSciNet  MATH  Google Scholar 

  • [H11] S. Helgason, Geometric Analysis on Symmetric Spaces, Amer. Math. Soc. (1994), Math. Surveys and Monographs.

    Google Scholar 

  • [H12] S. Helgason, Integral geometry and multitemporal wave equations, Comm. Pure Appl. Math ((to appear)).

    Google Scholar 

  • [HS] S. Helgason and H. Schlichtkrull, The Paley-Wiener space for the multitem-poral wave equation, Comm. Pure Appl. Math. ((to appear)).

    Google Scholar 

  • [I] S. Ishikawa. The range characterizations of the totally geodesic Radon transform on the real hyperbolic space, Preprint, Univ. of Tokyo, 1995.

    Google Scholar 

  • [IM] A. Intissar et M. Val Ould Moustapha, Solution explicite de l’équation des ondes dans l’espace symétrique de type non compact de rang 1, C.R. Acad. Sci. Paris 321 (1995), 77–80.

    Google Scholar 

  • [K] A. Kurusa, Support theorems for the totally geodesic Radon transform on constant curvature spaces, Proc. Amer. Math. Soc., 122 (1994), 429–435.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [L] D. Ludwig, The Radon transform on Euclidean space, Comm. Pure. Appl. Math. 23 (1966), 49–81.

    MathSciNet  MATH  Google Scholar 

  • [LP1] P. Lax and R.S. Phillips, Scattering Theory, Academic Press, New York, 1967.

    MATH  Google Scholar 

  • [LP2]-, Translation representations for the solution of the non-Euclidean wave equation, Comm. Pure. Appl. Math. 32 (1979), 617–667.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [O] B. Ørsted, The conformal invariance of Huygens’ principle, J. Differential Geom. 16 (1981), 1–9.

    MathSciNet  MATH  Google Scholar 

  • [Or] J. Orloff, Orbital integrals on symmetric spaces, in ‘Noncommutative Harmonic Analysis and Lie Groups’, Lecture Notes in Math. 1243 (1987), 198–239.

    CrossRef  MathSciNet  Google Scholar 

  • [P] E. Pedon, Équations des ondes sur les espaces hyperboliques, Preprint, Univ. de Nancy (1992–93).

    Google Scholar 

  • [PS] R.S. Phillips and M. Shahshahani, Scattering theory for symmetric spaces of the noncompact type, Duke Math. J. 72 (1993), 1–29.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [S] M. Shahshahani, Invariant hyperbolic systems on symmetric spaces, Differential Geometry (R. Brooks et al., eds.), Birkhäuser, boston, 1983, pp. 203–233.

    Google Scholar 

  • [So] D.C. Solomon, Asymptotic formulas for the dual Radon transform, Math. Zeitschr. 195 (1987), 321–343.

    CrossRef  MathSciNet  Google Scholar 

  • [SS] R. Schimming and H. Schlichtkrull, Helmholtz operators on harmonic manifolds, Acta Math. 173 (1994), 235–258.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [STS] M.A. Semenov-Tian-Shanski, Harmonic analysis on Riemannian symmetric spaces of negative curvature and scattering theory, Math. USSR Izvestija 10 (1976), 535–563.

    CrossRef  Google Scholar 

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© 1998 Springer-Verlag

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Helgason, S. (1998). Radon transforms and wave equations. In: Casadio Tarabusi, E., Picardello, M.A., Zampieri, G. (eds) Integral Geometry, Radon Transforms and Complex Analysis. Lecture Notes in Mathematics, vol 1684. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096092

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  • DOI: https://doi.org/10.1007/BFb0096092

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