Skip to main content
SpringerLink
Log in

A Note on the Strong Convergence of Partial Sums with Respect to Vilenkin System

  • Real and Complex Analysis
  • Published:
Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) Aims and scope Submit manuscript

Abstract

In this paper we prove some strong convergence theorems for partial sums with respect to Vilenkin system.

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. G. N. Agaev, N. Ya. Vilenkin, G. M. Dzafarly, A. I. Rubinstein, Multiplicative systems of functions and harmonic analysis on zero-dimensional groups (Ehim, Baku, 1981).

    Google Scholar 

  2. I. Blahota, K. Nagy, L.E. Persson, G. Tephnadze, “A sharp boundedness result concerning some maximal operators of partial sums with respect to Vilenkin systems”, Georgian Math. J., (to appear).

  3. I. Blahota, L. E. Persson, G. Tephnadze, “Two-sided estimates of the Lebesgue constants with respect to Vilenkin systems and applications”, Glasgow Math. J., doi: https://doi.org/10.1017/S0017089516000549.

    Article  MathSciNet  Google Scholar 

  4. R. R. Coifman, G.L. Weiss, “Extentions of Hardy spaces and their use in analysis”, Bull. Amer. Math. Acad. Sinica, 83, 569–645, 1977.

    Article  Google Scholar 

  5. S. Fridli, P. Simon, “On the Dirichlet kernels and Hardy space with resect to the Vilenkin system”, Acta Math. Hung., 45(1–2), 223–234, 1985.

    Article  Google Scholar 

  6. N. J. Fujii, “A maximal inequality for H 1 functions on the generalized Walsh-Paley group”, Proc. Amer. Math. Soc., 77, 111–116, 1979.

    MathSciNet  MATH  Google Scholar 

  7. G. Gát, “Investigations of certain operators with respect to the Vilenkin sistem”, Acta Math. Hung., 61, 131–149, 1993.

    Article  Google Scholar 

  8. B. I. Golubov, A.V. Efimov, V.A. Skvortsov, Walsh series and transforms (Kluwer, Dordrecht, 1991).

    Book  Google Scholar 

  9. S. F. Lukomskii, Lebesgue constants for characters of the compact zero-dimensional abelian group”, East J. Approx., 15(2), 219–231, 2009.

    MathSciNet  MATH  Google Scholar 

  10. I. Mezo, P. Simon, Integrals of weighted maximal kernels with respect to Vilenkin systems”, Publ. Math. Debrecen, 3580, 1–9, 2007.

    MathSciNet  MATH  Google Scholar 

  11. N. Memic, I. Simon, G. Tephnadze, “Strong convergence of two-dimensional Vilenkin-Fourier series”, Math. Nachr., 289(4), 485–500, 2016.

    Article  MathSciNet  Google Scholar 

  12. L. E. Persson, G. Tephnadze, P. Wall, “On an approximation of two-dimensional Walsh-Fourier series in the martingale Hardy spaces”, Ann. Funct. Anal., 9(1), 137–150, 2018.

    Article  MathSciNet  Google Scholar 

  13. L. E. Persson, G. Tephnadze, P. Wall, “On the maximal operators of Vilenkin-NЎrlund means”, J. Fourier Anal. Appl., 21(1), 76–94, 2015.

    Article  MathSciNet  Google Scholar 

  14. F. Schipp, W. R. Wade, P. Simon and J. Pál, Walsh series. An introduction to dyadic harmonic analysis (Adam Hilger, Bristol, 1990).

    MATH  Google Scholar 

  15. F. Schipp, “Certain rearrangements of series in the Walsh series”, Mat. Zametki, 18, 193–201, 1875.

    Google Scholar 

  16. P. Simon, “Investigations wish respect to the Vilenkin sistem”, Annales Univ. Sci. Budapest Eotv., Sect. Math., 28, 87–101, 1985.

    Google Scholar 

  17. P. Simon, “Strong convergence Theorem for Vilenkin-Fourier Series”, Journal of Mathematical Analysis and Applications, 245, 52–68, 2000.

    Article  MathSciNet  Google Scholar 

  18. P. Simon, “Strong Convergence Theorem for Vilenkin-Fourier Series”, J. Math. Anal. Appl., 245, 52–68, 2000.

    Article  MathSciNet  Google Scholar 

  19. P. Simon, “(L 1, H)-type estimations for some operators with respect to the Walsh? Paley system”, Acta Math. Hung., 46(3–4), 307–310, 1985.

    Article  Google Scholar 

  20. B. Smith, “A strong convergence theorem for H 1(T)”, Lecture Notes in Math., 995, 69–173, 1994.

    Google Scholar 

  21. G. Tephnadze, “On the partial sums of Walsh-Fourier series”, Colloq. Math., 141(2), 227–242, 2015.

    Article  MathSciNet  Google Scholar 

  22. G. Tephnadze, “A note on the norm convergence by Vilenkin-Fejér means”, Georgian Math. J., 21(4), 511–517, 2014.

    Article  MathSciNet  Google Scholar 

  23. G. Tephnadze, “On the partial sums of Vilenkin-Fourier series”, J. Contemp. Math. Anal., 49(1), 23–32, 2014.

    Article  MathSciNet  Google Scholar 

  24. G. Tephnadze, “Strong convergence of two-dimensional Walsh-Fourier series”, Ukr. Math. J., 65(6), 822–834, 2013.

    Article  MathSciNet  Google Scholar 

  25. N. Ya. Vilenkin, “A class of complete orthonormal systems”, Izv. Akad. Nauk SSSR, Ser. Mat., 11, 363–400, 1947.

    MathSciNet  MATH  Google Scholar 

  26. F. Weisz, Martingale Hardy spaces and their applications in Fourier Analysis, (Springer, Berlin, 1994).

    Book  Google Scholar 

  27. F. Weisz, Summability of multi-dimensional Fourier series and Hardy spaces. Mathematics and its Applications, 541 (Kluwer Academic Publishers, Dordrecht, 2002).

    Book  Google Scholar 

Download references

Acknowledgments

The author would like to thank the referee for helpful suggestions, which improved the final version of the paper.

Author information

Authors and Affiliations

  1. The University of Georgia, Tbilisi, Georgia

    G. Tutberidze

  2. Luleå University of Technology, Luleå, Sweden

    G. Tutberidze

Authors
  1. G. Tutberidze
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Tutberidze.

Additional information

Russian Text © The Author(s), 2019, published in Izvestiya Natsional’noi Akademii Nauk Armenii, Matematika, 2019, No. 6, pp. 81–87.

The research is supported by Shota Rustaveli National Science Foundation grant no. PHDF-18-476

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tutberidze, G. A Note on the Strong Convergence of Partial Sums with Respect to Vilenkin System. J. Contemp. Mathemat. Anal. 54, 365–370 (2019). https://doi.org/10.3103/S1068362319060062

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1068362319060062

MSC2010 numbers

Keywords

Search

Navigation