Ukrainian Mathematical Journal

, Volume 66, Issue 10, pp 1491–1508 | Cite as

On One Minkowski–Radon Problem and Its Generalizations

  • Vit. V. Volchkov
  • I. M. Savost’yanova

We study functions on a sphere with zero weighted means over the circles of fixed radius. A description of these functions is obtained in the form of series in special functions.


Entire Function Symmetric Space Integral Geometry Convolution Equation Invariant Differential Operator 
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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  1. 1.
    H. Minkowski, “Uber die Körper konstanter Breite,” Mat. Sb., 25, 505–508 (1904) (in Russian); Ges. Abh., 2, 277–279 (1904).Google Scholar
  2. 2.
    P. Funk, “Uber Flächen mit lauter geschlossenen geodätishen Linien,” Math. Ann., 74, 278–300 (1913).MathSciNetCrossRefGoogle Scholar
  3. 3.
    T. Bonnesen and W. Fenchel, Theorie der Konvexen Körper, Springer, Berlin (1934).CrossRefGoogle Scholar
  4. 4.
    C. M. Petty, “Centroid surfaces,” Pacif. J. Math., 11, 1535–1547 (1961).MathSciNetCrossRefGoogle Scholar
  5. 5.
    J. Radon, “Uber die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Ber. Ver. Sächs. Akad. Wiss. Leipzig, Math.-Nat. Kl., 69, 262–277 (1917).Google Scholar
  6. 6.
    P. Ungar, “Freak theorem about functions on a sphere,” J. London Math. Soc., 29, 100–103 (1954).MathSciNetCrossRefGoogle Scholar
  7. 7.
    R. Schneider, “Functions on a sphere and vanishing integrals over certain subspheres,” J. Math. Anal. Appl., 26, 381–384 (1969).MathSciNetCrossRefGoogle Scholar
  8. 8.
    R. Schneider, “Uber eine Integralgleichung in der Theorie der konvexen Körper,” Math. Nachr., 44, 55–75 (1970).MathSciNetCrossRefGoogle Scholar
  9. 9.
    C. A. Berenstein and L. Zalcman, “Pompeiu’s problem on spaces of constant curvature,” J. Anal. Math., 30, 113–130 (1976).MathSciNetCrossRefGoogle Scholar
  10. 10.
    C. A. Berenstein and L. Zalcman, “Pompeiu’s problem on symmetric spaces,” Comment. Math. Helv., 55, 593–621 (1980).MathSciNetCrossRefGoogle Scholar
  11. 11.
    E. Badertscher, “The Pompeiu problem on locally symmetric spaces,” J. Anal. Math., 57, 250–281 (1991).MathSciNetGoogle Scholar
  12. 12.
    V. V. Volchkov, Integral Geometry and Convolution Equations, Kluwer, Dordrecht (2003).CrossRefGoogle Scholar
  13. 13.
    V. V. Volchkov and Vit. V. Volchkov, Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group, Springer, London (2009).CrossRefGoogle Scholar
  14. 14.
    V. V. Volchkov and Vit. V. Volchkov, Offbeat Integral Geometry on Symmetric Spaces, Birkhäuser, Basel (2013).CrossRefGoogle Scholar
  15. 15.
    H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York (1954).Google Scholar
  16. 16.
    V. V. Volchkov, “Mean-value theorems for one class of polynomials,” Sib. Mat. Zh., 35, No. 4, 737–745 (1994).MathSciNetCrossRefGoogle Scholar
  17. 17.
    É. Ya. Riekstyn’sh, Asymptotic Expansions of Integrals [in Russian], Vol. 1, Zinatne, Riga (1974).Google Scholar
  18. 18.
    N. Ya. Vilenkin, Special Functions and the Theory of Representations of Groups [in Russian], Nauka, Moscow (1991).CrossRefGoogle Scholar
  19. 19.
    E. C. Titchmarsh, The Theory of Functions, Oxford University Press, New York (1939).Google Scholar
  20. 20.
    A. F. Leont’ev, Entire Functions. Series of Exponents [in Russian], Nauka, Moscow (1983).Google Scholar
  21. 21.
    S. Helgason, Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators and Spherical Functions, Academic Press, New York (1984).Google Scholar
  22. 22.
    Vit. V. Volchkov and I. M. Savost’yanova, “Analog of the John theorem for weighted spherical means on a sphere,” Ukr. Mat. Zh., 65, No. 5, 611–619 (2013); English translation: Ukr. Math. J., 65, No. 5, 674–683 (2013).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Vit. V. Volchkov
    • 1
  • I. M. Savost’yanova
    • 1
  1. 1.Donetsk National UniversityDonetskUkraine

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