Skip to main content
SpringerLink
Log in

On the injectivity of the Pompeiu transform for integral ball means

  • Published:
Ukrainian Mathematical Journal Aims and scope

A uniqueness theorem is proved for functions defined in \( {{\mathbb R}^n}, \; {n \geq 2} \), with vanishing integrals over the balls of fixed radius and a given majorant of growth. The problem of unimprovability of this theorem is analyzed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. V. Volchkov, Integral Geometry and Convolution Equations, Kluwer, Dordrecht (2003).

    Book  MATH  Google Scholar 

  2. V. V. Volchkov and Vit. V. Volchkov, Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group, Springer, London (2009).

    Book  MATH  Google Scholar 

  3. L. Zalcman, “Supplementary bibliography to ‘A bibliographic survey of the Pompeiu problem,”’ in: Radon Transforms and Tomography, American Mathematical Society, Providence, RI (2001), pp. 69–74.

    Google Scholar 

  4. I. D. Smith, “Harmonic analysis of scalar and vector fields in \( {{\mathbb R}^n} \);” Proc. Cambridge Phil. Soc., 72, 403–416 (1972).

    Article  MATH  Google Scholar 

  5. A. Sitaram, “Fourier analysis and determining sets for Radon measures on \( {{\mathbb R}^n} \);” Ill. J. Math., 28, No. 2, 339–347 (1984).

    MathSciNet  MATH  Google Scholar 

  6. S. Thangavelu, “Spherical means and CR functions on the Heisenberg group,” J. Anal. Math., 63, 255–286 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  7. V. V. Volchkov, “Solution of the problem of support for several classes of functions,” Mat. Sb., 188, No. 9, 13–30 (1997).

    MathSciNet  Google Scholar 

  8. M. Shahshahani and A. Sitaram, “The Pompeiu problem in exterior domains in symmetric space,” Contemp. Math., 63, 267–277 (1987).

    MathSciNet  Google Scholar 

  9. V. V. Volchkov, “Theorems on ball means in symmetric spaces,” Mat. Sb., 192, No. 9, 17–38 (2001).

    MathSciNet  Google Scholar 

  10. O. A. Ochakovskaya, “On functions with vanishing integrals over the balls of fixed radius in a half space,” Dokl. Ros. Akad. Nauk, 381, No. 6, 745–747 (2001).

    MathSciNet  Google Scholar 

  11. O. A. Ochakovskaya, “On functions with vanishing integrals over the balls of fixed radius,” Mat. Fiz., Anal., Geom., 9, No. 3, 493–501 (2002).

    MathSciNet  MATH  Google Scholar 

  12. O. A. Ochakovskaya, “Exact characteristics of the admissible rate of decay of nonzero functions with vanishing ball means,” Mat. Sb., 199, No. 1, 47–66 (2008).

    MathSciNet  Google Scholar 

  13. E. Seneta, Regularly Varying Functions, Springer, Berlin (1976).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences, Donetsk, Ukraine

    O. A. Ochakovskaya

Authors
  1. O. A. Ochakovskaya
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 3, pp. 361–368, March, 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ochakovskaya, O.A. On the injectivity of the Pompeiu transform for integral ball means. Ukr Math J 63, 416–424 (2011). https://doi.org/10.1007/s11253-011-0512-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-011-0512-1

Keywords

Search

Navigation