# zbMATH — the first resource for mathematics

Norm convergence of logarithmic means on unbounded Vilenkin groups. (English) Zbl 1394.42019
It is proved that if a sequence $$(m_n)$$ satisfies the condition $\sup_{n\in\mathbb{N}}\;\frac{\ln^2 m_1+\dots+\ln^2 m_n}{\ln m_1+\dots+\ln m_n} <\infty,$ then for the Vilenkin system generated by the sequence $$(m_n)$$ the convergence is guaranteed of the Riesz logarithmic means in the spaces $$L$$ and $$C$$. Consequently, an example is found of an unbounded Vilenkin system for which there exists a linear summation method providing the summability of the Fourier-Vilenkin series in the spaces $$L$$ and $$C$$.

##### MSC:
 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
Full Text:
##### References:
 [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. · Zbl 0595.42012 [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. · Zbl 1274.43006 [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. · Zbl 1164.42022 [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. · Zbl 0036.03604 [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. · Zbl 1129.42411 [11] U. Goginava, On the uniform convergence of Walsh-Fourier series, Acta Math. Hungar. 93 (2001), no. 1-2, 59-70. · Zbl 0992.42012 [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. · Zbl 1029.42022 [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. · Zbl 0345.42011 [14] J. Price, Certain groups of orthonormal step functions, Canad. J. Math. 9 (1957), 413-425. · Zbl 0079.09204 [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. · JFM 49.0707.01 [16] F. Schipp, On $$L_{p}$$-norm convergence of series with respect to product systems, Anal. Math. 2 (1976), no. 1, 49-64. · Zbl 0364.40009 [17] F. Schipp, W. R. Wade, P. Simon, Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol, 1990. · Zbl 0727.42017 [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. · Zbl 0060.18101 [20] F. T. Wang, On the summability of Fourier series by Riesz’s logarithmic means, II, Tohoku Math. J. 40 (1935), 392-397. · Zbl 0011.34601 [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. · Zbl 0192.42601 [22] W.-S. Young, Mean convergence of generalized Walsh-Fourier series, Trans. Amer. Math. Soc. 218 (1976), 311-320. · Zbl 0327.43009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.