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Strong convergence theorems for Walsh–Fejér means

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Abstract

As main result we prove that Fejér means of Walsh–Fourier series are uniformly bounded operators from H p to H p (0<p≦1/2).

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References

  1. I. Blahota, G. Gát and U. Goginava, Maximal operators of Fejér means of double Vilenkin–Fourier series, Colloq. Math., 107 (2007), 287–296.

    Article  MATH  MathSciNet  Google Scholar 

  2. I. Blahota, G. Gát and U. Goginava, Maximal operators of Fejér means of Vilenkin–Fourier series, J. Inequal. Pure Appl. Math., 7 (2006), 1–7.

    Google Scholar 

  3. N. J. Fine, On the Walsh function, Trans. Amer. Math. Soc., 65 (1949), 372–414.

    Article  MATH  MathSciNet  Google Scholar 

  4. N. J. Fujii, A maximal inequality for H 1 functions on the generalized Walsh–Paley group, Proc. Amer. Math. Soc., 77 (1979), 111–116.

    MATH  MathSciNet  Google Scholar 

  5. G. Gát, Investigations of certain operators with respect to the Vilenkin system, Acta Math. Hungar., 61 (1993), 131–149.

    Article  MATH  MathSciNet  Google Scholar 

  6. G. Gát, Cesàro means of integrable functions with respect to unbounded Vilenkin systems, J. Approx. Theory, 124 (2003), 25–43.

    Article  MATH  MathSciNet  Google Scholar 

  7. U. Goginava, Maximal operators of Fejér means of double Walsh–Fourier series, Acta Math. Hungar., 115 (2007), 333–340.

    Article  MATH  MathSciNet  Google Scholar 

  8. U. Goginava, On the approximation properties of Cesàro means of negative order of Walsh–Fourier series, J. Approx. Theory, 115 (2002), 9–20.

    Article  MATH  MathSciNet  Google Scholar 

  9. U. Goginava, Maximal operators of Fejér–Walsh means, Acta Sci. Math. (Szeged), 74 (2008), 615–624.

    MATH  MathSciNet  Google Scholar 

  10. U. Goginava, The martingale Hardy type inequality for Marcinkiewicz–Fejér means of two-dimensional conjugate Walsh–Fourier series, Acta Math. Sinica, 27 (2011), 1949–1958.

    Article  MATH  MathSciNet  Google Scholar 

  11. B. Golubov, A. Efimov and V. Skvortsov, Walsh Series and Transformations, Kluwer Acad. Publ. (Dordrecht, Boston, London, 1991).

    Book  Google Scholar 

  12. J. Pál and P. Simon, On a generalization of the comncept of derivate, Acta Math. Hungar., 29 (1977), 155–164.

    Article  MATH  Google Scholar 

  13. F. Schipp, W. R. Wade, P. Simon and J. Pál, Walsh Series, an Introduction to Diadic Harmonic Analysis, Akadémiai Kiadó, AdamHilger (Budapest–Bristol–New York, 1990).

    Google Scholar 

  14. F. Schipp, Certain rearranngements of series in the Walsh series, Mat. Zametki, 18 (1975), 193–201.

    MATH  MathSciNet  Google Scholar 

  15. P. Simon, Investigations with respect to the Vilenkin system, Annales Univ. Sci. Budapest. Eötvös, Sect. Math., 28 (1985), 87–101.

    Google Scholar 

  16. P. Simon, A note on the of the Sunouchi operator with respect to Vilenkin systems, Annales Univ. Sci. Sect. Math. Budapest (2001).

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. P. Simon and F. Weisz, Strong convergence theorem for two-parameter Vilenkin–Fourier series, Acta Math. Hungar., 86 (2000), 17–38.

    Article  MATH  MathSciNet  Google Scholar 

  20. B. Smith, A strong convergence theorem for H 1(T), in: Lecture Notes in Math. 995, Springer (Berlin, 1994), pp. 169–173.

    Google Scholar 

  21. G. Tephnadze, Fejér means of Vilenkin–Fourier series, Studia Sci. Math. Hung., 49 (2012), 79–90.

    MATH  MathSciNet  Google Scholar 

  22. G. Tephnadze, On the maximal operator of Vilenkin–Fejér means, Turk. J. Math., 37 (2013), 308–318.

    MATH  MathSciNet  Google Scholar 

  23. F. Weisz, Hardy spaces and Cesáro means of two-dimensional Fourier series, Bolyai Soc. Math. Studies (1996), 353–367.

  24. F. Weisz, Strong convergence theorems for two-parameter Walsh–Fourier and trigonometric-Fourier series, Studia Math., 117 (1996), 173–194.

    MATH  MathSciNet  Google Scholar 

  25. F. Weisz, Martingale Hardy Spaces and their Applications in Fourier Analysis, Springer (Berlin–Heideiberg–New York, 1994).

    MATH  Google Scholar 

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

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Exact and Natural Sciences, Tbilisi State University, Chavchavadze str. 1, Tbilisi, 0128, Georgia

    George Tephnadze

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  1. George Tephnadze
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Correspondence to George Tephnadze.

Additional information

The research was supported by Shota Rustaveli National Science Foundation grant no. 13/06 (Geometry of function spaces, interpolation and embedding theorems).

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Tephnadze, G. Strong convergence theorems for Walsh–Fejér means. Acta Math Hung 142, 244–259 (2014). https://doi.org/10.1007/s10474-013-0361-5

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  • DOI: https://doi.org/10.1007/s10474-013-0361-5

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