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

DOI:
https://doi.org/10.1090/S0002-9939-02-06554-1

Published electronically:
May 8, 2002

MathSciNet review:
1908925

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be either the hyperbolic space or the unit sphere , and let be the set of all -dimensional totally geodesic submanifolds of . For and , the totally geodesic Radon transform is studied. By averaging over all at a distance from , and applying Riemann-Liouville fractional differentiation in , S. Helgason has recovered . We show that in the hyperbolic case this method blows up if 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 , are obtained.

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Additional Information

**Boris Rubin**

Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel

Email:
boris@math.huji.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-02-06554-1

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