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Journal of Mathematical Sciences

, Volume 158, Issue 2, pp 235–240 | Cite as

Inversion of the Kipriyanov–Radon transform via fractional derivatives in a one-dimensional parameter

  • L. N. LyakhovEmail author
  • G. Gots
Article

Abstract

This paper considers the Kipriyanov–Radon transform constructed as a special Radon transform adopted for dealing with singular Bessel differential operators of the corresponding indices acting on a part of the variables. The authors obtain inversion formulas generalizing the classical formulas for the Radon transform of axially-symmetric functions and relating to the integro-differentiation of fractional order in a one-dimensional parameter.

Keywords

Radon Fractional Order Fractional Derivative Inversion Formula Integral Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    I. M. Gel’fand, M. I. Graev, and N. Ya. Vilenkin, Integral Geometry and Representation Theory Problems Related to It [in Russian], GIFML, Moscow (1962).Google Scholar
  2. 2.
    E. G. Gots and L. N. Lyakhov, “Inversion of the Kipriyanov –Radon transform by using the Grünwald–Letnikov–Riesz fractional differentiation,” Dokl. Ross. Akad. Nauk, 412, No. 1, 1–4 (2007).MathSciNetGoogle Scholar
  3. 3.
    S. Helgason, Groups and Geometric Analysis [Russian translation], Mir, Moscow (1987).Google Scholar
  4. 4.
    I. A. Kipriyanov and L. N. Lyakhov, “On Fourier, Fourier–Bessel, and Radon transforms,” Dokl. Ross. Akad. Nauk, 360, No. 2, 157–160 (1998).zbMATHMathSciNetGoogle Scholar
  5. 5.
    I. A. Kipriyanov, Singular Elliptic Boundary-Value Problems [in Russian], Nauka, Moscow (1997).zbMATHGoogle Scholar
  6. 6.
    L. N. Lyakhov, “Kipriyanov–Radon transform,” Trudy Mat. Inst. Ross. Akad. Nauk, 248, 153–163 (2005).MathSciNetGoogle Scholar
  7. 7.
    L. N. Lyakhov, “Inversion of the Kipriyanov–Radon transform,” Dokl. Ross. Akad. Nauk, 399, No. 5, 597–600 (2004).MathSciNetGoogle Scholar
  8. 8.
    S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals of Fractional Order and Certain Their Applications [in Russian], Nauka i Tekhnika, Minsk (1987).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.Voronezh State Technological AcademyVoronezhRussia

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