Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 44A12, 52A22

Retrieve articles in all journals with MSC (2000): 44A12, 52A22

Additional Information

Boris Rubin
Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
MR Author ID: 209987

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