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Helgason-Marchaud inversion formulas for Radon transforms

Author: Boris Rubin
Journal: Proc. Amer. Math. Soc. 130 (2002), 3017-3023
MSC (2000): Primary 44A12; Secondary 52A22
Published electronically: May 8, 2002
MathSciNet review: 1908925
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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

Keywords: Geodesic Radon transforms, Marchaud's fractional derivatives
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
Article copyright: © Copyright 2002 American Mathematical Society

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