Norm summability of Nörlund logarithmic means on unbounded Vilenkin groups. (English) Zbl 1164.42022

Summary: The (Nörlund) logarithmic means of the Fourier series is: \(t_nf=\frac 1{l_n}\sum\limits^{n-1}_{k=1}\frac{S_kf}{n-k}\), where \(l_n=\sum\limits^{n-1}_{k=1}\frac 1k\). In general, the Fejér \((C, 1)\) means have better properties than the logarithmic ones. We compare them and show that in the case of some unbounded Vilenkin systems the situation changes.


42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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[1] Agaev, G. H., Ya, N., Vilenkin, G. M. and Dzhafarli, A. I., Rubinstein, Multiplicative Systems of Functions and Harmonic Analysis on 0-Dimensional Groups, Izd. ”ELM” (Baku, USSR), (1981).
[2] Gát, G., Cesaro Means of Integrable Functions with Respect to Unbounded Vilenkin Systems, Journal of Approximation Theory, 124:1(2003), 25–43. · Zbl 1032.43003 · doi:10.1016/S0021-9045(03)00075-3
[3] Gát, G., Pointwise Convergence of the Fejér Means of Functions on Unbounded Vilenkin Groups, Journal of Approximation Theory, 101:1(1999), 1–36. · Zbl 0972.42019 · doi:10.1006/jath.1999.3346
[4] Gát, G. and Goginava, U., Uniform and L-Convergence of Logarithmic Means of Walsh-Fourier Series, Acta Mathematica Sinica (English Series), 22:2(2006), 497–506. · Zbl 1129.42411 · doi:10.1007/s10114-005-0648-8
[5] Goginava, U. and Tkebuchava, G., Convergence of Subsequences of Partial Sums and Logarithmic Means of Walsh-Fourier Series, Acta Sci. Math. (Szeged), 70(1992), 159–177. · Zbl 1109.42008
[6] Móricz, F., Approximation by Nörlund Means of Walsh-Fourier Series, Journal of Approximation Theory, 70(1992), 375–389. · Zbl 0757.42009 · doi:10.1016/0021-9045(92)90067-X
[7] Price, J., Certain Groups of Orthonormal Step Functions, Canadian J. Math., 9(1957), 413–425. · Zbl 0079.09204 · doi:10.4153/CJM-1957-049-x
[8] Simon, P., Verallgemeinerte Walsh-Fourierreihen II., Acta Mathematica Acad. Sci. Hungar., 27(1976), 329–341. · Zbl 0335.42009 · doi:10.1007/BF01902112
[9] Simon, P. and Pál, J., On a Generalization of the Concept Derivative, Acta Mathematica Acad. Sci. Hungar., 29(1977), 155–164. · Zbl 0345.42011 · doi:10.1007/BF01896477
[10] Vilenkin, N. Ya., A Class of Complete Orthonormal Systems, Izv. Akad. Nauk. U.S.S.R., Ser. Mat., 11(1947), 363–400. · Zbl 0036.35601
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