Skip to main content
Log in

Strong Convergence of Two-Dimensional Walsh–Fourier Series

  • Published:
Ukrainian Mathematical Journal Aims and scope

We prove that certain means of quadratic partial sums of the two-dimensional Walsh–Fourier series are uniformly bounded operators acting from the Hardy space H p to the space L p for 0 < p < 1.

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

Access this article

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. Gát, “Investigations of certain operators with respect to the Vilenkin system,” Acta Math. Hung., 61, 131–149 (1993).

    Article  MATH  Google Scholar 

  2. G. Gát, U. Goginava, and K. Nagy, “On the Marcinkiewicz–Fejár means of double Fourier series with respect to theWalsh–Kaczmarz system,” Stud. Sci. Math. Hung., 46, No. 3, 399–421 (2009).

    MATH  Google Scholar 

  3. G. Gát, U. Goginava, and G. Tkebuchava, “Convergence in measure of logarithmic means of quadratical partial sums of double Walsh–Fourier series,” J. Math. Anal. Appl., 323, No. 1, 535–549 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  4. U. Goginava, “The weak type inequality for the maximal operator of the Marcinkiewicz–Fejer means of the two-dimensionalWalsh–Fourier series,” J. Approxim. Theory, 154, No. 2, 161–180 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  5. U. Goginava and L. D. Gogoladze, “Strong convergence of cubic partial sums of two-dimensional Walsh–Fourier series,” Constructive Theory of Functions (Sozopol, 2010): In the Memory of Borislav Bojanov, Acad. Publ. House, Sofia (2012), pp. 108–117.

  6. L. D. Gogoladze, “On the strong summability of Fourier series,” Bull Acad. Sci. Georg. SSR, 52, No. 2, 287–292 (1968).

    MathSciNet  MATH  Google Scholar 

  7. B. Golubov, A. Efimov, and V. Skvortsov, Walsh Series and Transformations, Kluwer Acad. Publ., Dordrecht, etc. (1991).

  8. F. Schipp, W. R. Wade, P. Simon, and J. Pál, Walsh Series. Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol, New York (1990).

  9. P. Simon, “Strong convergence of certain means with respect to the Walsh–Fourier series,” Acta Math. Hung., 49, 425–431 (1987).

    Article  MATH  Google Scholar 

  10. P. Simon, “Strong convergence theorem for Vilenkin–Fourier series,” J. Math. Anal. Appl., 245, 52–68 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  11. B. Smith, “A strong convergence theorem for H 1(T),” Lect. Notes Math., 995, 169–173 (1994).

    Article  Google Scholar 

  12. G. Tephnadze, “On the Vilenkin–Fourier coefficients,” Georg. Math. J. (to appear).

  13. G. Tephnadze, “A note of the Fourier coefficients and partial sums of Vilenkin–Fourier series,” Acta Math. Acad. Ped. Nyiregyhaziensis (to appear).

  14. F. Weisz, Martingale Hardy Spaces and Their Applications in Fourier Analysis, Springer, Berlin etc. (1994).

  15. F. Weisz, Summability of Multi-Dimensional Fourier Series and Hardy Space, Kluwer Acad. Publ., Dordrecht, etc. (2002).

  16. F. Weisz, “Strong convergence theorems for two-parameter Walsh–Fourier and trigonometric-Fourier series,” Stud. Math., 117, No. 2, 173–194 (1996).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 6, pp. 822–834, June, 2013.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tephnadze, G. Strong Convergence of Two-Dimensional Walsh–Fourier Series. Ukr Math J 65, 914–927 (2013). https://doi.org/10.1007/s11253-013-0828-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-013-0828-0

Keywords

Navigation