Skip to main content
SpringerLink
Log in

Weighted Integrability of \(p\)-Adic Fourier Transform

  • RESEARCH ARTICLES
  • Published:
p-Adic Numbers, Ultrametric Analysis and Applications Aims and scope Submit manuscript

Abstract

We obtain sufficient conditions for functions defined on \(p\)-adic linear space providing the weighted integrability of their Fourier transforms. The Bernstein-Szasz type conditions connected with moduli of smoothness are sharp in a certain sense. As a corollary we deduce recent results of S. S. Platonov. Also we prove Zygmund type tests for integrability of functions having bounded \(s\)-fluctuation and belonging to a Hölder class.

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. S. Vladimirov, I. V. Volovich and E.I. Zelenov, \(p\)-Adic Analysis and Mathematical Physics (World Scientific, Singapore, 1994).

    Book  Google Scholar 

  2. N. Koblitz, \(p\)-Adic Numbers, \(p\)-Adic Analysis, and Zeta Functions (Springer-Verlag, N.Y., 1984).

    Book  Google Scholar 

  3. M. H. Taibleson, Fourier Analysis on Local Fields (Princeton Univ. Press, Princeton, 1975).

    MATH  Google Scholar 

  4. L. Gogoladze, R. Meskhia, “On the absolute convergence of trigonometric Fourier series,” Proc. Razmazde Math. Inst. 141, 29–40 (2006).

    MathSciNet  MATH  Google Scholar 

  5. F. Móricz, “Sufficient conditions for the Lebesgue integrability of Fourier transforms,” Anal. Math. 36 (2), 121–129 (2010).

    Article  MathSciNet  Google Scholar 

  6. S. S. Platonov, “Fourier transform of Dini-Lipschitz functions on the field of \(p\)-adic numbers,” \(p\)-Adic Numbers Ultrametric Anal. Appl. 11 (4), 307–318(2019).

    Article  MathSciNet  Google Scholar 

  7. B. I. Golubov and S. S. Volosivets, “On the integrability and uniform convergence of multiplicative Fourier transform,” Georgian Math. J. 16 (3), 533–546 (2009).

    MathSciNet  MATH  Google Scholar 

  8. B. I. Golubov and S. S. Volosivets, “Generalized weighted integrability of the multiplicative Fourier transform,” Proceedings of MIPT 3 (1), 49–56 (2011) [in Russian].

    Google Scholar 

  9. S. S. Volosivets and M. A. Kuznetsova, “Generalized \(p\)-adic Fourier transform and estimates of integral modulus of continuity in terms of this transform,” \(p\)-Adic Numbers Ultrametric Anal. Appl. 10 (4), 312–321 (2018).

    Article  MathSciNet  Google Scholar 

  10. S. S. Volosivets, “Hausdorff operators on \(p\)-adic linear spaces and their properties in Hardy, BMO, and Hölder spaces,” Math. Notes 93, 382–391 (2013).

    Article  MathSciNet  Google Scholar 

  11. A. Zygmund, Trigonometric Series, Vol.1 (Cambridge Univ. Press, Cambridge, 1959).

    MATH  Google Scholar 

Download references

Funding

The work of second author was supported by the Ministry of Science and Education of the Russian Federation in the framework of the basic part of the scientific research state task, project FSRR-2020-0006 .

Author information

Authors and Affiliations

  1. Moscow Institute of Physics and Technology (State University), Moscow, Russia

    Boris I. Golubov

  2. Faculty of Mechanics and Mathematics, Saratov State University, Saratov, Russia

    Sergey S. Volosivets

Authors
  1. Boris I. Golubov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergey S. Volosivets
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergey S. Volosivets.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Golubov, B.I., Volosivets, S.S. Weighted Integrability of \(p\)-Adic Fourier Transform. P-Adic Num Ultrametr Anal Appl 12, 203–209 (2020). https://doi.org/10.1134/S2070046620030036

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S2070046620030036

Keywords

Search

Navigation