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Cesàro and Riesz summability with varying parameters of multi-dimensional Walsh–Fourier series

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Abstract

It is proved that the maximal operator of subsequences of the Cesàro and Riesz means with varying parameters is bounded from the dyadic Hardy space Hp to Lp. This implies an almost everywhere convergence for the subsequences of tqoo.

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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), 59–78

  2. P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Birkhäuser (Basel, 1971)

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

    Article  MathSciNet  Google Scholar 

  4. Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc. 83, 569–645 (1977)

    Article  MathSciNet  Google Scholar 

  5. Fefferman, R.: Calderon-Zygmund theory for product domains: \(H^p\) spaces. Proc. Nat. Acad. Sci. USA 83, 840–843 (1986)

    Article  Google Scholar 

  6. Fejér, L.: Untersuchungen über Fouriersche Reihen. Math. Ann. 58, 51–69 (1904)

    Article  Google Scholar 

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

    Article  Google Scholar 

  8. Fujii, N.: A maximal inequality for \({H}^1\)-functions on a generalized Walsh-Paley group. Proc. Amer. Math. Soc. 77, 111–116 (1979)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  11. L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education (New Jersey, 2004)

  12. Herz, C.: Bounded mean oscillation and regulated martingales. Trans. Amer. Math. Soc. 193, 199–215 (1974)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Lebesgue, H.: Recherches sur la convergence des séries de Fourier. Math. Ann. 61, 251–280 (1905)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  16. Riesz, M.: Sur la sommation des séries de Fourier. Acta Sci. Math. (Szeged) 1, 104–113 (1923)

    MATH  Google Scholar 

  17. Schipp, F.: Über gewissen Maximaloperatoren. Ann. Univ. Sci. Budapest Sect. Math. 18, 189–195 (1975)

    MathSciNet  MATH  Google Scholar 

  18. F. Schipp and P. Simon, On some \(({H},{L}_1)\)-type maximal inequalities with respect to the Walsh–Paley system, in: Functions, Series, Operators, Proc. Conf. in Budapest, 1980, Coll. Math. Soc. J. Bolyai, vol. 35, North Holland (Amsterdam, 1981), pp. 1039–1045

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. 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 

  22. Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Univ, Press (Princeton, N.J. (1971)

    MATH  Google Scholar 

  23. Trigub, R.M., Belinsky, E.S.: Fourier Analysis and Approximation of Functions, Kluwer Academic Publishers. Boston, London, Dordrecht (2004)

    Book  Google Scholar 

  24. F. Weisz, Martingale Hardy Spaces and their Applications in Fourier Analysis, Lecture Notes in Math., vol. 1568, Springer (Berlin, 1994)

  25. Weisz, F.: Cesàro summability of two-parameter Walsh-Fourier series. J. Approx. Theory 88, 168–192 (1997)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. Weisz, F.: Summability of Multi-dimensional Fourier Series and Hardy Spaces, Mathematics and Its Applications, Kluwer Academic Publishers. Boston, London, Dordrecht (2002)

    Google Scholar 

  28. Weisz, F.: Summability of multi-dimensional trigonometric Fourier series. Surv. Approx. Theory 7, 1–179 (2012)

    MathSciNet  MATH  Google Scholar 

  29. Weisz, F.: Convergence and Summability of Fourier Transforms and Hardy Spaces. Applied and Numerical Harmonic Analysis, Springer, Birkhäuser (Basel (2017)

    Book  Google Scholar 

  30. A. Zygmund, Trigonometric Series (3rd ed.), Cambridge Press (London, 2002)

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Authors and Affiliations

  1. Department of Numerical Analysis, Eötvös L. University, Pázmány P. sétány 1/C, H-1117, Budapest, Hungary

    F. Weisz

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  1. F. Weisz
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Correspondence to F. Weisz.

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This research was supported by the Hungarian National Research, Development and Innovation Office – NKFIH, K115804 and KH130426.

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Weisz, F. Cesàro and Riesz summability with varying parameters of multi-dimensional Walsh–Fourier series. Acta Math. Hungar. 161, 292–312 (2020). https://doi.org/10.1007/s10474-020-01024-7

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  • DOI: https://doi.org/10.1007/s10474-020-01024-7

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