HelgasonMarchaud inversion formulas for Radon transforms
Author:
Boris Rubin
Journal:
Proc. Amer. Math. Soc. 130 (2002), 30173023
MSC (2000):
Primary 44A12; Secondary 52A22
Published electronically:
May 8, 2002
MathSciNet review:
1908925
Fulltext PDF Free Access
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 RiemannLiouville 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 RiemannLiouville ones. New inversion formulas for , are obtained.
 [1]
C.A. Berenstein, E. Casadio Tarabusi, Inversion formulas for the dimensional Radon transform in real hyperbolic spaces, Duke Math. Journal 62 (1991), 613631. MR 93b:53056
 [2]
C.A. Berenstein, B. Rubin, Radon transform of functions on the Lobachevsky space and hyperbolic wavelet transforms, Forum Math. 11 (1999), 567590. MR 2000e:44001
 [3]
, Totally geodesic Radon transform of functions on real hyperbolic space, Preprint, 2001.
 [4]
S. Helgason, Differential operators on homogeneous spaces, Acta Math. 102 (1959), 239299. MR 22:8457
 [5]
, The totally geodesic Radon transform on constant curvature spaces, Contemp. Math 113 (1990), 141149. 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), 641650. 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), 202215. MR 99c:44003
 [12]
R.S. Strichartz, estimates for Radon transforms in Euclidean and noneuclidean spaces, Duke Math. J. 48 (1981), 699727. MR 86k:43008
 [1]
 C.A. Berenstein, E. Casadio Tarabusi, Inversion formulas for the dimensional Radon transform in real hyperbolic spaces, Duke Math. Journal 62 (1991), 613631. MR 93b:53056
 [2]
 C.A. Berenstein, B. Rubin, Radon transform of functions on the Lobachevsky space and hyperbolic wavelet transforms, Forum Math. 11 (1999), 567590. MR 2000e:44001
 [3]
 , Totally geodesic Radon transform of functions on real hyperbolic space, Preprint, 2001.
 [4]
 S. Helgason, Differential operators on homogeneous spaces, Acta Math. 102 (1959), 239299. MR 22:8457
 [5]
 , The totally geodesic Radon transform on constant curvature spaces, Contemp. Math 113 (1990), 141149. 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), 641650. 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), 202215. MR 99c:44003
 [12]
 R.S. Strichartz, estimates for Radon transforms in Euclidean and noneuclidean spaces, Duke Math. J. 48 (1981), 699727. 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/S0002993902065541
PII:
S 00029939(02)065541
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
