On the approximation properties of Cesàro means of negative order of double Vilenkin – Fourier series

  • T. Tepnadze I. Javakhishvili Tbilisi State Univ., Georgia


UDC 517.5

We establish approximation properties of Ces\`{a}ro $% (C,-\alpha ,-\beta)$ means with $\alpha ,\beta $ $\epsilon $ $(0,1)$ of Vilenkin\,--\,Fourier series. This result allows one to obtain a condition which is sufficient for the convergence of the means $\sigma _{n,m}^{-\alpha,-\beta }(x,y,f)$ to $f(x,y)$ in the $L^{p}$-metric.


Agaev, G. N.; Vilenkin, N. Ya.; Dzhafarli, G. M.; Rubinshteĭn, A. I. Мультипликативные системы функций и гармонический анализ на нульмерных группах. (Russian) [[Multiplicative systems of functions and harmonic analysis on zero-dimensional groups]] ``Èlm'', Baku, 1981. 180 pp. MR0679132

Fine, N. J. Cesàro summability of Walsh-Fourier series. Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 588--591. doi: 10.1073/pnas.41.8.588

Golubov, B. I.; Efimov, A. V.; Skvortsov, V. A. Ряды и преобразования Уолша. (Russian) [[Walsh series and transforms]] Теория и применения. [Theory and applications] ``Nauka'', Moscow, 1987. 344 pp. MR0925004

Goginava, U. On the uniform convergence of Walsh-Fourier series. Acta Math. Hungar. 93 (2001), no. 1-2, 59--70. doi: 10.1023/A:1013865315680

Goginava, Ushangi. On the approximation properties of Cesàro means of negative order of Walsh-Fourier series. J. Approx. Theory 115 (2002), no. 1, 9--20. doi: 10.1006/jath.2001.3632

Goginava, Ushangi. Uniform convergence of Cesàro means of negative order of double Walsh-Fourier series. J. Approx. Theory 124 (2003), no. 1, 96--108. doi: 10.1016/S0021-9045(03)00134-5

Goginava, Ushangi. Cesàro means of double Walsh-Fourier series. Anal. Math. 30 (2004), no. 4, 289--304. doi: 10.1007/s10476-005-0516-x

Goginava, Ushangi; Nagy, Károly. On the maximal operator of Walsh-Kaczmarz-Fejér means. Czechoslovak Math. J. 61(136) (2011), no. 3, 673--686. doi: 10.1007/s10587-011-0038-6

Gát, Gy.; Goginava, U. A weak type inequality for the maximal operator of $(C,alpha)$-means of Fourier series with respect to the Walsh-Kaczmarz system. Acta Math. Hungar. 125 (2009), no. 1-2, 65--83. doi: 10.1007/s10474-009-8217-8

Gát, G.; Nagy, K. Cesàro summability of the character system of the $p$-series field in the Kaczmarz rearrangement. Anal. Math. 28 (2002), no. 1, 1--23. doi: 10.1023/A:1014893314662

Nagy, Károly. Approximation by Cesàro means of negative order of Walsh-Kaczmarz-Fourier series. East J. Approx. 16 (2010), no. 3, 297--311. MR2789336

Simon, Péter; Weisz, Ferenc. Weak inequalities for Cesàro and Riesz summability of Walsh-Fourier series. J. Approx. Theory 151 (2008), no. 1, 1--19. doi: 10.1016/j.jat.2007.05.004

Schipp, F. Über gewisse Maximaloperatoren. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 18 (1975), 189--195 (1976). MR0430665

Schipp, F.; Wade, W. R.; Simon, P. Walsh series. An introduction to dyadic harmonic analysis. With the collaboration of J. Pál. Adam Hilger, Ltd., Bristol, 1990. {rm x}+560 pp. ISBN: 0-7503-0068-X MR1117682

Tepnadze, Tsitsino. On the approximation properties of Cesàro means of negative order of Vilenkin-Fourier series. Studia Sci. Math. Hungar. 53 (2016), no. 4, 532--544. doi: 10.1556/012.2016.53.4.1350

Tevzadze, V. Uniform $(C,alpha)(-1 < alpha < 0)$ summability of Fourier series with respect to the Walsh-Paley system. Acta Math. Acad. Paedagog. Nyházi. (N.S.) 22 (2006), no. 1, 41--61. MR2216766

Zhizhiashvili, Levan. Trigonometric Fourier series and their conjugates. Revised and updated translation of Some problems of the theory of trigonometric Fourier series and their conjugate series (Russian) [Tbilis. Gos. Univ., Tbilisi, 1993]. Translated from the Russian by George Kvinikadze. Mathematics and its Applications, 372. Kluwer Academic Publishers Group, Dordrecht, 1996. {rm xii}+300 pp. ISBN: 0-7923-4088-4 doi: 10.1007/978-94-009-0283-1

Zygmund, A. Trigonometric series: Vols. I, II. Second edition, reprinted with corrections and some additions Cambridge University Press, London-New York 1968 Vol. I. {rm xiv}+383 pp.; Vol. II: {rm vii}+364 pp. (two volumes bound as one). MR0236587

How to Cite
Tepnadze, T. “On the Approximation Properties of Cesàro Means of Negative Order of Double Vilenkin – Fourier Series”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 3, Mar. 2020, pp. 391-06, doi:10.37863/umzh.v72i3.6045.
Research articles