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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A Radon transform on spheres through the origin in $\textbf {R}^{n}$ and applications to the Darboux equation
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by A. M. Cormack and E. T. Quinto PDF
Trans. Amer. Math. Soc. 260 (1980), 575-581 Request permission

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
  • © Copyright 1980 American Mathematical Society
  • 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