We establish approximation properties of Cesàro (C,−α,−β) means with α, β 𝜖 (0, 1) for the Vilenkin–Fourier series. This result enables one to establish a condition sufficient for the convergence of the means \( {\sigma}_{n,m}^{-\alpha, -\beta } \) (x, y, f) to f(x, y) in the Lp-metric.
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References
G. N. Agaev, N. Ya. Vilenkin, G. M. Dzhafarli, and A. I. Rubinshtein, Multiplicative Systems of Functions and Harmonic Analysis on Zero-Dimensional Groups [in Russian], Ehlm, Baku (1981).
N. J. Fine, “Cesàro summability of Walsh–Fourier series,” Proc. Nat. Acad. Sci. USA, 41, 558–591 (1995).
B. I. Golubov, A.V. Efimov, and V. A. Skvortsov, Walsh Series and Transformations. Theory and Applications [in Russian], Nauka, Moscow (1987).
U. Goginava, “On the uniform convergence of Walsh–Fourier series,” Acta Math. Hungar., 93, No. 1-2, 59–70 (2001).
U. Goginava, “On the approximation properties of Cesàro means of negative order of Walsh–Fourier series,” J. Approx. Theory, 115, No. 1, 9–20 (2002).
U. Goginava, “Uniform convergence of Cesàro means of negative order of double Walsh–Fourier series,” J. Approx. Theory, 124, No. 1, 96–108 (2003).
U. Goginava, “Cesàro means of double Walsh–Fourier series,” Anal. Math., 30, 289–304 (2004).
U. Goginava and K. Nagy, “On the maximal operator of Walsh–Kaczmarz–Fejér means,” Czechoslovak Math. J., 61(136), No. 3, 673–686 (2011).
G. Gát and U. Goginava, “A weak type inequality for the maximal operator of (C, α)-means of Fourier series with respect to the Walsh–Kaczmarz system,” Acta Math. Hungar., 125, No. 1-2, 65–83 (2009).
G. Gát and K. Nagy, “Cesàro summability of the character system of the p-series field in the Kaczmarz rearrangement,” Anal. Math., 28, No. 1, 1–23 (2002).
K. Nagy, “Approximation by Cesàro means of negative order ofWalsh–Kaczmarz–Fourier series,” East J. Approx., 16, No. 3, 297–311 (2010).
P. Simon and F.Weisz, “Weak inequalities for Cesàro and Riesz summability ofWalsh–Fourier series,” J. Approx. Theory, 151, No. 1, 1–19 (2008).
F. Schipp, “Über gewisse Maximaloperatoren,” Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 18, 189–195 (1975).
F. Schipp, W. R. Wade, P. Simon, and J. Pál, Walsh Series, Introduction to Dyadic Harmonic Analysis, Hilger, Bristol (1990).
T. Tepnadze, “On the approximation properties of Cesàro means of negative order of Vilenkin–Fourier series,” Studia Sci. Math. Hungar., 53, No. 4, 532–544 (2016).
V. I. Tevzadze, “Uniform (C,−α) summability of Fourier series with respect to theWalsh–Paley system,” Acta Math. Acad. Paedagog. Nyházi. (N.S.), 22, No. 1, 41–61 (2006).
L. V. Zhizhiashvili, Trigonometric Fourier Series and Their Conjugates [in Russian], Tbilisi (1993).
A. Zygmund, Trigonometric Series, Vo1. 1, Cambridge Univ. Press, Cambridge, UK (1959).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 3, pp. 391–406, March, 2020.
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Tepnadze, T. On the Approximation Properties of Cesàro Means of Negative Order for the Double Vilenkin–Fourier Series. Ukr Math J 72, 446–463 (2020). https://doi.org/10.1007/s11253-020-01792-z
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DOI: https://doi.org/10.1007/s11253-020-01792-z