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Transactions of the American Mathematical Society

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A Radon transform on spheres through the origin in $ {\bf R}\sp{n}$ and applications to the Darboux equation


Authors: A. M. Cormack and E. T. Quinto
Journal: Trans. Amer. Math. Soc. 260 (1980), 575-581
MSC: Primary 44A05; Secondary 33A45, 35Q05, 43A55, 58G15
DOI: https://doi.org/10.1090/S0002-9947-1980-0574800-3
MathSciNet review: 574800
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Abstract: On domain $ {C^\infty }\,({R^n})$ we invert the Radon transform that maps a function to its mean values on spheres containing the origin. Our inversion formula implies that if $ f\, \in \,{C^\infty }\,({R^n})$ and its transform is zero on spheres inside a disc centered at 0, then f is zero inside that disc. We give functions $ f\, \notin \,{C^\infty }\,({R^n})$ whose transforms are identically zero and we give a necessary condition for a function to be the transform of a rapidly decreasing function. We show that every entire function is the transform of a real analytic function. These results imply that smooth solutions to the classical Darboux equation are determined by the data on any characteristic cone with vertex on the initial surface; if the data is zero near the vertex then so is the solution. If the data is entire then a real analytic solution with that data exists.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0574800-3
Keywords: Radon transform, spherical means, Darboux partial differential equation, Gegenbauer polynomials, spherical harmonics
Article copyright: © Copyright 1980 American Mathematical Society

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