Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
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?)

  • [1] C.A. Berenstein, E. Casadio Tarabusi, Inversion formulas for the $k$-dimensional Radon transform in real hyperbolic spaces, Duke Math. Journal 62 (1991), 613-631. MR 93b:53056
  • [2] C.A. Berenstein, B. Rubin, Radon transform of $L^{p}$-functions on the Lobachevsky space and hyperbolic wavelet transforms, Forum Math. 11 (1999), 567-590. MR 2000e:44001
  • [3] -, Totally geodesic Radon transform of $L^{p}$-functions on real hyperbolic space, Preprint, 2001.
  • [4] S. Helgason, Differential operators on homogeneous spaces, Acta Math. 102 (1959), 239-299. MR 22:8457
  • [5] -, The totally geodesic Radon transform on constant curvature spaces, Contemp. Math 113 (1990), 141-149. MR 92j:53036
  • [6] -, The Radon transform, Birkhäuser, Boston, Second edition, 1999. MR 2000m:44003
  • [7] D. M. Oberlin, E. M. Stein, Mapping properties of the Radon transform, Indiana Univ. Math. J. 31 (1982), 641-650. MR 84a:44002
  • [8] B. Rubin, Fractional integrals and potentials, Pitman Monographs and Surveys in Pure and Applied Mathematics, 82, Longman, Harlow, 1996. MR 98h:42018
  • [9] -, Inversion formulas for the spherical Radon transform and the generalized cosine transform, Advances in Appl. Math. (to appear).
  • [10] -, Radon, cosine, and sine transforms on real hyperbolic space, Advances in Math. (to appear).
  • [11] -, Spherical Radon transforms and related wavelet transforms, Applied and Computational Harm. Anal. 5 (1998), 202-215. MR 99c:44003
  • [12] R.S. Strichartz, $L^{p}$-estimates for Radon transforms in Euclidean and non-euclidean spaces, Duke Math. J. 48 (1981), 699-727. MR 86k:43008

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
Email: boris@math.huji.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06554-1
PII: S 0002-9939(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