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Maximal operators of Cesàro means with varying parameters of Walsh–Fourier series

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Abstract

The boundedness of maximal operators of subsequences of \((C,\alpha _{n})\)-means of partial sums of Walsh–Fourier series from the Hardy space Hp into the space Lp is studied.

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References

  1. T. Akhobadze, On the convergence of generalized Cesàro means of trigonometric Fourier series. I, Acta Math. Hungar., 115 (2007), no. 1-2, 59–78

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Fine, J.: Cesàro summability of Walsh-Fourier series. Proc. Nat. Acad. Sci. USA 41, 558–591 (1955)

    Article  Google Scholar 

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

    MathSciNet  Google Scholar 

  5. Gát, G.: On the Fejér kernel functions with respect to the Walsh-Kaczmarz systems. Acta Acad. Paed. Agriensis Sectio Math. 24, 105–110 (1997)

    MATH  Google Scholar 

  6. A. A. Abu Joudeh and G. Gát, Convergence of Cesàro means with varying parameters of Walsh–Fourier series, Miskolc Math. Notes, 19 (2018), 303–317

  7. 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 

  8. Goginava, U.: The maximal operator of Marcinkiewicz-Fejér means of the \(d\)-dimensional Walsh-Fourier series. East J. Approx. 12, 295–302 (2006)

    MathSciNet  Google Scholar 

  9. Goginava, U.: The weak type inequality for the Walsh system. Studia Math. 185, 35–48 (2008)

    Article  MathSciNet  Google Scholar 

  10. Goginava, U., Weisz, F.: Maximal operator of the Fejér means of triangular partial sums of two-dimensional Walsh-Fourier series. Georgian Math. J. 19, 101–115 (2012)

    MathSciNet  MATH  Google Scholar 

  11. Marcinkiewicz, I., Zygmund, A.: On the summability of double Fourier series. Fund. Math. 32, 112–132 (1939)

    MATH  Google Scholar 

  12. Schipp, F.: Certain rearrangements of series in the Walsh series. Mat. Zametki 18, 193–201 (1975)

    MathSciNet  Google Scholar 

  13. Simon, P.: Investigation with respect to the Vilenkin system. Ann. Univ. Sci. Sect. Math. (Budapest) 27, 87–101 (1985)

    MathSciNet  MATH  Google Scholar 

  14. Simon, P.: Cesàro summability with respect to two-parameter Walsh system. Monatsh. Math. 131, 321–334 (2000)

    Article  MathSciNet  Google Scholar 

  15. Simon, P., Weisz, F.: Weak inequalities for Cesàro and Riesz summability of Walsh-Fourier series. J. Approx. Theory 151, 1–19 (2008)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  17. F. Weisz, Martingale Hardy Spaces and Their Applications in Fourier Analysis, Springer (Berlin–Heidelberg–New York, 1994)

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

    Article  MathSciNet  Google Scholar 

  19. Weisz, F.: \(( C,\alpha ) \) summability of Walsh-Fourier series. Anal. Math. 27, 141–156 (2001)

    Article  MathSciNet  Google Scholar 

  20. F. Weisz, Summability of Multi-dimensional Fourier Series and Hardy Space, Kluwer Academic (Dordrecht, 2002)

  21. Weisz, F.: Cesàro summability of two-dimensional Walsh-Fourier series. Trans. Amer. Math. Soc. 348, 2169–2181 (1996)

    Article  MathSciNet  Google Scholar 

  22. Zygmund, A.: Trigonometric Series, vol. 1. Cambridge Univ, Press (1959)

    MATH  Google Scholar 

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Acknowledgement

The authors are indebted to the anonymous referee for finding some error in the first version of the manuscript.

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Correspondence to G. Gát.

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The first author is supported by the Hungarian National Foundation for Scientific Research (OTKA), grant no. K111651 and by project EFOP-3.6.2-16-2017-00015 supported by the European Union, cofinanced by the European Social Fund.

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Gát, G., Goginava, U. Maximal operators of Cesàro means with varying parameters of Walsh–Fourier series. Acta Math. Hungar. 159, 653–668 (2019). https://doi.org/10.1007/s10474-019-00961-2

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  • DOI: https://doi.org/10.1007/s10474-019-00961-2

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