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Proceedings of the American Mathematical Society

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

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Helgason-Marchaud inversion formulas for Radon transforms
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by Boris Rubin PDF
Proc. Amer. Math. Soc. 130 (2002), 3017-3023 Request permission

Abstract:

Let $X$ be either the hyperbolic space $\mathbb {H} ^{n}$ or the unit sphere $S^{n}$, and let $\Xi$ be the set of all $k$-dimensional totally geodesic submanifolds of $X, 1 \le k \le n-1$. For $x \in X$ and $\xi \in \Xi$, the totally geodesic Radon transform $f(x) \to \hat f(\xi )$ is studied. By averaging $\hat f(\xi )$ over all $\xi$ at a distance $\theta$ from $x$, and applying Riemann-Liouville fractional differentiation in $\theta$, S. Helgason has recovered $f(x)$. We show that in the hyperbolic case this method blows up if $f$ does not decrease sufficiently fast. The situation can be saved if one employs Marchaud’s fractional derivatives instead of the Riemann-Liouville ones. New inversion formulas for $\hat f(\xi ), f \in L^{p}(X)$, are obtained.
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Additional Information
  • Boris Rubin
  • Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
  • MR Author ID: 209987
  • Email: boris@math.huji.ac.il
  • Received by editor(s): May 16, 2001
  • Published electronically: May 8, 2002
  • Additional Notes: This work was partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).
  • Communicated by: David Preiss
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 130 (2002), 3017-3023
  • MSC (2000): Primary 44A12; Secondary 52A22
  • DOI: https://doi.org/10.1090/S0002-9939-02-06554-1
  • MathSciNet review: 1908925