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Strong Convergence Theorem for Walsh–Kaczmarz–Fejér Means

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Abstract

As main result we prove that Fejér means of Walsh–Kaczmarz–Fourier series are uniformly bounded operators from the Hardy martingale space \(\ H_{p}\) to the Hardy martingale space \(H_{p}\) for \( 0<p\le 1/2\).

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References

  1. Agaev, G.N., Vilenkin, N.Ya., Dzhafarli, G.M., Rubinstein, A.I.: Multiplicative systems of functions and harmonic analysis on 0-dimensional groups, “ELM”. Baku, USSR) (1981). (Russian)

  2. Blahota, I.: On a norm inequality with respect to Vilenkin-like systems. Acta Math. Hungar. 89(1–2), 15–27 (2000)

    Article  MathSciNet  Google Scholar 

  3. Blahota, I.: On the maximal value of Dirichlet and Fejér kernels with respect to the Vilenkin-like space. Publ. Math. 80(3–4), 503–513 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Blahota, I., Nagy, K., Persson, L.E., Tephnadze, G.: A sharp boundedness result concerning some maximal operators of partial sums with respect to Vilenkin systems, Georgian Math. Georgian Math. J. 26(3), 351–360 (2019)

    Article  MathSciNet  Google Scholar 

  5. Gát, G.: On \((C,1)\) summability of integrable functions with respect to the Walsh–Kaczmarz system. Studia Math. 130(2), 135–148 (1998)

    Article  MathSciNet  Google Scholar 

  6. Gát, G.: Inverstigations of certain operators with respect to the Vilenkin sistem. Acta Math. Hung. 61, 131–149 (1993)

    Article  Google Scholar 

  7. Gát, G., Goginava, U., Nagy, K.: On the Marcinkiewicz–Fejér means of double Fourier series with respect to the Walsh-Kaczmarz system. Studia Sci. Math. Hung. 46(3), 399–421 (2009)

    MATH  Google Scholar 

  8. Goginava, U.: The maximal operator of the Fejér means of the character system of the \(p\)-series field in the Kaczmarz rearrangement. Publ. Math. Debrecen 71(1–2), 43–55 (2007)

    MathSciNet  MATH  Google Scholar 

  9. Goginava, U., Nagy, K.: On the maximal operator of Walsh–Kaczmarz-Fejér means. Czeh. Math. J. 61(136), 673–686 (2011)

    Article  Google Scholar 

  10. Golubov, B., Efimov, A., Skvortsov, V.: Walsh series and transformations, Dordrecht, Boston, London, 1991. Kluwer Acad. publ., (1991)

  11. Nagy, K., Tephnadze, G.: Kaczmarz–Marcinkiewicz means and Hardy spaces. Acta math. Hung. 149(2), 346–374 (2016)

    Article  MathSciNet  Google Scholar 

  12. Nagy, K., Tephnadze, G.: On the Walsh–Marcinkiewicz means on the Hardy space. Cent. Eur. J. Math. 12(8), 1214–1228 (2014)

    MathSciNet  MATH  Google Scholar 

  13. Nagy, K., Tephnadze, G.: Strong convergence theorem for Walsh–Marcinkiewicz means. Math. Inequal. Appl. 19(1), 185–195 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Persson, L.-E., Tephnadze, G.: A sharp boundedness result concerning some maximal operators of Vilenkin–Fejér means. Mediterr. J. Math. 13(4), 1841–1853 (2016)

    Article  MathSciNet  Google Scholar 

  15. Persson, L.-E., Tephnadze, G., Tutberidze, G.: On the boundedness of subsequences of Vilenkin-Fejér means on the martingale Hardy spaces, operators and matrices, 14 (1) (2020), 283-294

  16. Persson, L.-E., Tephnadze, G., Tutberidze, G., Wall, P.: Strong summability result of Vilenkin-Fejér means on bounded Vilenkin groups, Ukr. Math. J., (to appear)

  17. Schipp, F.: Pointwise convergence of expansions with respect to certain product systems. Anal. Math. 2, 65–76 (1976)

    Article  MathSciNet  Google Scholar 

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

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

    Article  MathSciNet  Google Scholar 

  20. Simon, P.: On the Cesàro summability with respect to the Walsh–Kaczmarz system. J. Approx. Theory 106, 249–261 (2000)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Simon, P.: \((C,\alpha )\) summability of Walsh–Kaczmarz–Fourier series. J. Approx. Theory 127, 39–60 (2004)

    Article  MathSciNet  Google Scholar 

  23. Simon, P.: Remarks on strong convergence with respect to the Walsh system. East J. Approx. 6, 261–276 (2000)

    MathSciNet  MATH  Google Scholar 

  24. Skvortsov, V.A.: On Fourier series with respect to the Walsh–Kaczmarz system. Anal. Math. 7, 141–150 (1981)

    Article  MathSciNet  Google Scholar 

  25. Smith, B.: A strong convergence theorem for \( H_{1} ( T ) ,\) Lecture Notes in Math., 995, Springer, Berlin, (1994), 169–173

  26. S̆neider, A.A.: On series with respect to the Walsh functions with monotone coefficients. Izv. Akad. Nauk SSSR Ser. Math 12, 179–192 (1948)

    Google Scholar 

  27. Tephnadze, G.: On the maximal operators of Walsh–Kaczmarz-Fejér means. Period. Math. Hungar. 67(1), 33–45 (2013)

    Article  MathSciNet  Google Scholar 

  28. Tephnadze, G.: Approximation by Walsh–Kaczmarz–Fejér means on the Hardy space, Acta Math. Scientia, (34B) (5) (2014) 1593–1602

  29. Tephnadze, G.: A note on the norm convergence by Vilenkin–Fejér means. Georgian Math. J. 21(4), 511–517 (2014)

    Article  MathSciNet  Google Scholar 

  30. Tephnadze, G.: Strong convergence theorem for Walsh–Fejér means. Acta Math. Hungar. 142(1), 244–259 (2014)

    Article  MathSciNet  Google Scholar 

  31. Tutberidze, G.: A note on the strong convergence of partial sums with respect to Vilenkin system. J. Contemp. Math. Anal. 54(6), 319–324 (2019)

    MathSciNet  MATH  Google Scholar 

  32. Young, W.S.: On the a.e converence of Walsh–Kaczmarz–Fourier series. Proc. Amer. Math. Soc 44, 353–358 (1974)

    Article  MathSciNet  Google Scholar 

  33. Weisz, F.: Martingale Hardy spaces and their applications in Fourier Analysis. Springer, Berlin-Heidelberg-New York (1994)

    Book  Google Scholar 

  34. Weisz, F.: Summability of multi-dimensional Fourier series and Hardy space. Kluwer Academic, Dordrecht (2002)

    Book  Google Scholar 

  35. Weisz, F.: \(\theta \)-summability of Fourier series. Acta Math. Hungar. 103, 139–176 (2004)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Science and Technology, The University of Georgia, 77a Merab Kostava St, Tbilisi, 0128, Georgia

    Nata Gogolashvili & George Tephnadze

  2. Institute of Mathematics and Computer Sciences, University of Nyíregyháza, P.O. Box 166, Nyíregyháza, 4400, Hungary

    Károly Nagy

Authors
  1. Nata Gogolashvili
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  2. Károly Nagy
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  3. George Tephnadze
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Correspondence to George Tephnadze.

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Research supported by projects TÁMOP-4.2.2.A-11/1/KONV-2012-0051, GINOP-2.2.1-15-2017-00055 and by Shota Rustaveli National Science Foundation grant no. FR-19-676.

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Gogolashvili, N., Nagy, K. & Tephnadze, G. Strong Convergence Theorem for Walsh–Kaczmarz–Fejér Means. Mediterr. J. Math. 18, 37 (2021). https://doi.org/10.1007/s00009-020-01682-5

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  • DOI: https://doi.org/10.1007/s00009-020-01682-5

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