Skip to main content
SpringerLink
Log in

Almost everywhere summability of two-dimensional walsh-fourier series

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

It is proved that the maximal operators of subsequences of Cesàro means with varying parameters of two-dimensional Walsh-Fourier series is bounded from the dyadic Hardy spaces \(H_{p}\left( {\mathbb {I}}\right) \) to \(L_{p}\left( {\mathbb {I}}\right) \). This implies an almost everywhere convergence for the subsequences of the summability means.

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. Akhobadze, T.: Lebesgue constants of generalized Cesáro \((C,\alpha _n)\)-means of trigonometric series. Bull. Georgian Natl. Acad. Sci. 175(2), 24–26 (2007)

    MathSciNet  MATH  Google Scholar 

  2. Akhobadze, T.: On the convergence of generalized Cesàro means of trigonometric Fourier series. II. Acta Math. Hungar. 115(1–2), 79–100 (2007)

    Article  MathSciNet  Google Scholar 

  3. Chang, S.-Y.A., Fefferman, R.: Some recent developments in Fourier analysis and \(H^p\)-theory on product domains. Bull. Amer. Math. Soc. (N.S.) 12(1), 1–43 (1985)

  4. Fefferman, R.: Calderón-Zygmund theory for product domains: \(H^p\) spaces. Proc. Nat. Acad. Sci. U.S.A. 83(4), 840–843 (1986)

    Article  MathSciNet  Google Scholar 

  5. Fine, J.: Cesaro summability of Walsh-Fourier series. Proc. Nat. Acad. Sci. United States of America 41(8), 588 (1955)

    Article  MathSciNet  Google Scholar 

  6. Fujii, N.J.: Cesáro summability of Walsh-Fourier series. In Proc. Amer. Math. Soc 77, 111–116 (1979)

    Google Scholar 

  7. Gát, G.: On the divergence of the \((C,1)\) means of double Walsh-Fourier series. Proc. Amer. Math. Soc. 128(6), 1711–1720 (2000)

    Article  MathSciNet  Google Scholar 

  8. Gát, G., Goginava, U.: Maximal operators of Cesàro means with varying parameters of Walsh-Fourier series. Acta Math. Hungar. 159(2), 653–668 (2019)

    Article  MathSciNet  Google Scholar 

  9. Gát, G., Goginava, U.: Almost everywhere convergence and divergence of cesàro means with varying parameters of walsh–fourier series. Arabian Journal of Mathematics, pp. 1–19 (2021)

  10. Goginava, U.: The maximal operator of the \((C,\alpha )\) means of the Walsh-Fourier series. Ann. Univ. Sci. Budapest. Sect. Comput. 26, 127–135 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Goginava, U.: Maximal \((C,\alpha ,\beta )\) operators of two-dimensional Walsh-Fourier series. Acta Math. Acad. Paedagog. Nyházi. (N.S.) 24(2), 209–214 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Goginava, U., Ben Said, S.: Maximal summability operators on the dyadic Hardy spaces. Filomat 35(7), 2189–2208 (2021)

    Article  MathSciNet  Google Scholar 

  13. Golubov, B., Efimov, A., Skvortsov, V.: Walsh series and transforms, volume 64 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991. Theory and applications, Translated from the 1987 Russian original by W. R. Wade

  14. Joudeh, A., Gát, G.: Convergence of Cesáro means with varying parameters of Walsh-Fourier series. Miskolc Math. Notes 19(1), 303–317 (2018)

    Article  MathSciNet  Google Scholar 

  15. Kaplan, I.B.: Cesàro means of variable order. Izv. Vysš. Učebn. Zaved. Matematika 5(18), 62–73 (1960)

    Google Scholar 

  16. Móricz, F., Schipp, F., Wade, W.R.: Cesàro summability of double Walsh-Fourier series. Trans. Amer. Math. Soc. 329(1), 131–140 (1992)

    MathSciNet  MATH  Google Scholar 

  17. Schipp, F.: On certain rearrangements of series with respect to the Walsh system. Mat. Zametki 18(2), 193–201 (1975)

    MathSciNet  Google Scholar 

  18. Schipp, F., Wade, W. R., Simon, P.: Walsh series. Adam Hilger, Ltd., Bristol, 1990. An introduction to dyadic harmonic analysis, With the collaboration of J. Pál

  19. Simon, P.: Investigations with respect to the Vilenkin system. Annales Univ. Sci. Budapestiensis, Sectio Math 27, 87–101 (1985)

    MathSciNet  MATH  Google Scholar 

  20. Tetunashvili, Sh.: On divergence of Fourier series by some methods of summability. J. Funct. Spaces Appl., pages Art. ID 542607, 9, (2012)

  21. Weisz, F.: Martingale Hardy spaces and their applications in Fourier analysis. Lecture Notes in Mathematics, vol. 1568. Springer-Verlag, Berlin (1994)

  22. Weisz, F.: Cesàro summability of one- and two-dimensional Walsh-Fourier series. Anal. Math. 22(3), 229–242 (1996)

    Article  MathSciNet  Google Scholar 

  23. Weisz, F.: The maximal \((C,\alpha,\beta )\) operator of two-parameter Walsh-Fourier series. J. Fourier Anal. Appl. 6(4), 389–401 (2000)

    Article  MathSciNet  Google Scholar 

  24. Weisz, F.: \((C,\alpha )\) summability of Walsh-Fourier series. Anal. Math. 27(2), 141–155 (2001)

    Article  MathSciNet  Google Scholar 

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

  26. Weisz, F.: Cesàro and Riesz summability with varying parameters of multi-dimensional Walsh-Fourier series. Acta Math. Hungar. 161(1), 292–312 (2020)

    Article  MathSciNet  Google Scholar 

  27. Weisz, F.: Summability of Fourier series in periodic Hardy spaces with variable exponent. Acta Math. Hungar. 162(2), 557–583 (2020)

    Article  MathSciNet  Google Scholar 

  28. Zhizhiashvili, L.: Trigonometric Fourier series and their conjugates, volume 372 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1996. Revised and updated translation of ıt Some problems of the theory of trigonometric Fourier series and their conjugate series (Russian) [Tbilis. Gos. Univ., Tbilisi, 1993], Translated from the Russian by George Kvinikadze

  29. Zygmund, A.: Trigonometric series. Vol. I, II. Cambridge Mathematical Library. Cambridge University Press, Cambridge, third edition, 2002. With a foreword by Robert A. Fefferman

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, United Arab Emirates University, P.O. Box No. 15551, Al Ain, Abu Dhabi, UAE

    Ushangi Goginava

Authors
  1. Ushangi Goginava
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ushangi Goginava.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors would like to thankfully acknowledge the financial support awarded by UAEU through the Start-up Grant No. 12S100.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Goginava, U. Almost everywhere summability of two-dimensional walsh-fourier series. Positivity 26, 63 (2022). https://doi.org/10.1007/s11117-022-00925-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11117-022-00925-x

Keywords

Mathematics Subject Classification

Search

Navigation