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Transactions of the American Mathematical Society

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A representation of the solutions of the Darboux equation in odd-dimensional spaces


Author: H. Rhee
Journal: Trans. Amer. Math. Soc. 150 (1970), 491-498
MSC: Primary 35.06
MathSciNet review: 0262647
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Abstract: It is shown that determining a function from its averages over all spheres passing through the origin leads to an explicit representation of the even solutions of the Darboux equation in the exterior of the characteristic cones in terms of the hyperboloidal means of the boundary data on the cones.


References [Enhancements On Off] (What's this?)

  • [1] Y. W. Chen, On the solutions of the wave equation in the exterior of the characteristic cones, J. Math. Mech. 16 (1967), 655–673. MR 0212418
  • [2] Fritz John, Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, 1955. MR 0075429
  • [3] Fritz John, Bestimmung einer Funktion aus ihren Integralen Über gewisse Mannigfaltigkeiten, Math. Ann. 109 (1934), no. 1, 488–520 (German). MR 1512906, 10.1007/BF01449151
  • [4] H. Rhee, Inversion of the functional equation $ {\text{SM} ^ \ast }f = J$ in odd-dimensional spaces, Thesis, Univ. of Massachusetts, Amherst, 1968.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0262647-0
Keywords: Darboux equation, characteristic cones, spherical means, hyperboloidal means, John-Asgeiersson identity, Radon transform
Article copyright: © Copyright 1970 American Mathematical Society