Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A generalization of the Mader-Helgason inversion formulas for Radon transforms

Authors: Y. A. Antipov and B. Rubin
Journal: Trans. Amer. Math. Soc. 364 (2012), 6479-6493
MSC (2010): Primary 44A12; Secondary 47G10
Published electronically: June 12, 2012
MathSciNet review: 2958944
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In 1927, Philomena Mader derived elegant inversion formulas for the hyperplane Radon transform on $ \mathbb{R}^n$. These formulas differ from the original ones by Radon and seem to be forgotten. We generalize Mader's formulas to totally geodesic Radon transforms in any dimension on arbitrary constant curvature space. Another new interesting inversion formula for the $ k$-plane transform was presented in the recent book ``Integral geometry and Radon transform'' by S. Helgason. We extend this formula to arbitrary constant curvature space. The paper combines tools of integral geometry and complex analysis.

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

  • 1. C. A. Berenstein and B. Rubin, Totally geodesic Radon transform of $ L^p$-functions on real hyperbolic space. Fourier analysis and convexity, 37-58, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2004. MR 2087237 (2005f:44003)
  • 2. S.R. Deans, The Radon transform and some of its applications, Dover Publ. Inc., Mineola, New York, 2007. MR 1274701 (95a:44003)
  • 3. I.M. Gel'fand, S.G. Gindikin and M.I. Graev, Selected topics in integral geometry, Translations of Mathematical Monographs, Amer. Math. Soc., Providence, RI, 2003. MR 2000133 (2004f:53092)
  • 4. S. Helgason, Integral geometry and Radon transform, Springer, New York-Dordrecht-Heidelberg-London, 2011. MR 2743116
  • 5. Ph. Mader, Über die Darstellung von Punktfunktionen im n-dimensionalen euklidischen Raum durch Ebenenintegrale, Math. Zeit. 26 (1927), 646-652. MR 1544879
  • 6. J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math. - Nat. Kl., 69 (1917), 262-277 (Russian translation is in the Russian edition of S. Helgason, The Radon transform, Moscow, Mir, 1983, pp. 134-148; English translation is in [2]). MR 692055 (84f:01040)
  • 7. B. Rubin, Helgason-Marchaud inversion formulas for Radon transforms, Proc. Amer. Math. Soc. 130 (2002), 3017-3023. MR 1908925 (2003f:44003)
  • 8. B. Rubin, Inversion formulas for the spherical Radon transform and the generalized cosine transform. Advances in Appl. Math. 29 (2002), 471-497. MR 1942635 (2004c:44006)
  • 9. B. Rubin, Reconstruction of functions from their integrals over $ k$-dimensional planes, Israel J. of Math. 141 (2004), 93-117. MR 2063027 (2005b:44004)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 44A12, 47G10

Retrieve articles in all journals with MSC (2010): 44A12, 47G10

Additional Information

Y. A. Antipov
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803

B. Rubin
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803

Keywords: Radon transforms, inversion formulas, constant curvature space.
Received by editor(s): March 10, 2011
Published electronically: June 12, 2012
Additional Notes: The first author was supported by the NSF grant DMS-0707724.
The second author was supported by the NSF grants PFUND-137 (Louisiana Board of Regents) and DMS-0556157.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society