Banach Journal of Mathematical Analysis

Norm convergence of logarithmic means on unbounded Vilenkin groups

György Gát and Ushangi Goginava

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In this paper we prove that, in the case of some unbounded Vilenkin groups, the Riesz logarithmic means converges in the norm of the spaces X(G) for every fX(G), where by X(G) we denote either the class of continuous functions with supremum norm or the class of integrable functions.

Article information

Banach J. Math. Anal., Volume 12, Number 2 (2018), 422-438.

Received: 28 March 2017
Accepted: 17 July 2017
First available in Project Euclid: 17 November 2017

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Zentralblatt MATH identifier

Primary: 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
Secondary: 40G99: None of the above, but in this section

unbounded Vilenkin group Riesz logarithmic means convergence in norm


Gát, György; Goginava, Ushangi. Norm convergence of logarithmic means on unbounded Vilenkin groups. Banach J. Math. Anal. 12 (2018), no. 2, 422--438. doi:10.1215/17358787-2017-0031.

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  • [1] G. N. Agaev, N. Ya. Vilenkin, G. M. Dzhafarli, and A. I. Rubinshteĭn, Multiplicative Systems of Functions and Harmonic Analysis on Zero-dimensional Groups, Ehlm, Baku, 1981.
  • [2] M. Avdispahić, Concepts of generalized bounded variation and the theory of Fourier series, Int. J. Math. Math. Sci. 9 (1986), no. 2, 223–244.
  • [3] M. Avdispahić and N. Memić, On the Lebesgue test for convergence of Fourier series on unbounded Vilenkin groups, Acta Math. Hungar. 129 (2010), no. 4, 381–392.
  • [4] M. Avdispahić and M. Pepić, On summability in $L_{p}$-norm on general Vilenkin groups, Taiwanese J. Math. 4 (2000), no. 2, 285–296.
  • [5] M. Avdispahić and M. Pepić, Summability and integrability of Vilenkin series, Collect. Math. 51 (2000), no. 3, 237–254.
  • [6] I. Blahota and G. Gát, Norm summability of Nörlund logarithmic means on unbounded Vilenkin groups, Anal. Theory Appl. 24 (2008), no. 1, 1–17.
  • [7] A. V. Efimov, On certain approximation properties of periodic multiplicative orthonormal systems, Mat. Sb. (N.S.) 69 (1966), 354–370.
  • [8] N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372–414.
  • [9] G. Gát, Cesàro means of integrable functions with respect to unbounded Vilenkin systems, J. Approx. Theory 124 (2003), no. 1, 25–43.
  • [10] G. Gát and U. Goginava, Uniform and $L$-convergence of logarithmic means of Walsh-Fourier series, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 2, 497–506.
  • [11] U. Goginava, On the uniform convergence of Walsh-Fourier series, Acta Math. Hungar. 93 (2001), no. 1–2, 59–70.
  • [12] U. Goginava, Uniform convergence of Cesàro means of negative order of double Walsh-Fourier series, J. Approx. Theory 124 (2003), no. 1, 96–108.
  • [13] J. Pál and P. Simon, On a generalization of the concept of derivative, Acta Math. Acad. Sci. Hungar. 29 (1977), no. 1–2, 155–164.
  • [14] J. Price, Certain groups of orthonormal step functions, Canad. J. Math. 9 (1957), 413–425.
  • [15] M. Riesz, Sur un théorème de la moyenne et ses applications, Acta Litt. ac Scient. Univ. Hung. 1 (1923), 114–126. Acta. Sci. Math. (Szeged), 1 (1922), 114-126.
  • [16] F. Schipp, On $L_{p}$-norm convergence of series with respect to product systems, Anal. Math. 2 (1976), no. 1, 49–64.
  • [17] F. Schipp, W. R. Wade, P. Simon, Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol, 1990.
  • [18] P. Simon, Verallgemeinerte Walsh-Fourierreihen, II. Acta Math. Acad. Sci. Hungar. 27 (1976), no. 3–4, 329–341.
  • [19] O. Szász, On the logarithmic means of rearranged partial sums of Fourier series, Bull. Amer. Math. Soc. 48 (1942), 705–711.
  • [20] F. T. Wang, On the summability of Fourier series by Riesz’s logarithmic means, II, Tohoku Math. J. 40 (1935), 392–397.
  • [21] K. Yabuta, Quasi-Tauberian theorems, applied to the summability of Fourier series by Riesz’s logarithmic means, Tohoku Math. J. (2) 22 (1970), 117–129.
  • [22] W.-S. Young, Mean convergence of generalized Walsh-Fourier series, Trans. Amer. Math. Soc. 218 (1976), 311–320.