On the approximation properties of Cesàro means of negative order of Walsh-Fourier series. (English) Zbl 0998.42018

Some approximation properties of Cesàro \((C,-\alpha)\) means with \(0<\alpha<1\) of Walsh-Fourier series are given. A sufficient condition for the convergence of the means \(\sigma_n^{-\alpha} f\) to \(f\) in \(L_p\) norm \((1\leq p \leq \infty)\) is obtained. It is also shown that this condition cannot be improved in the case \(p=1\).


42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42B08 Summability in several variables
Full Text: DOI


[1] Fine, N. J., Cesàro summability of Walsh-Fourier series, Proc. Nat. Acad. Sci. U.S.A., 41, 558-591 (1995) · Zbl 0065.05303
[2] Glukhov, V. A., On the summability of multiple Fourier series with respect to multiplicative systems, Mat. Zametki, 39, 665-673 (1986) · Zbl 0607.42020
[3] Golubov, B. I.; Efimov, A. V.; Skvortsov, V. A., Series and transformations of Walsh (1987), Nauka: Nauka Moscow · Zbl 0692.42009
[4] Goginava, U., On the convergence and summability of \(N\)-dimensional Fourier series with respect to the Walsh-Paley systems in the spaces \(L^p\)([0, 1]\(^N), p\)∈[1, ∞], Georgian Math. J., 7, 1-22 (2000) · Zbl 0976.42015
[5] Goginava, U., Convergence and summability of multiple Fourier-Walsh series in \(L^p\)([0, 1]\(^N)\) metrics, Bull. Georgian Acad. Sci., 158, 11-13 (1998) · Zbl 0918.42020
[6] Paley, A., A remarkable series of orthogonal functions, Proc. London Math. Soc., 34, 241-279 (1992) · JFM 58.0284.03
[7] Schipp, F.; Wade, W. R.; Simon, P.; Pàl, J., Walsh Series, Introduction to Dyadic Harmonic Analysis (1990), Hilger: Hilger Bristol · Zbl 0727.42017
[8] Schipp, F., Über gewisse Maximaloperatoren, Ann. Univ. Sci. Budapest. Sect. Math., 18, 189-195 (1975) · Zbl 0351.42012
[9] Tevzadze, V. I., Uniform convergence of Cesàro means of negative order of Fourier-Walsh series, Soobshch. Akad. Nauk Gruzii SSR, 102, 33-36 (1981) · Zbl 0466.42004
[10] Zhizhiashvili, L. V., Trigonometric Fourier Series and Their Conjugates (1993)
[11] Zygmund, A., Trigonometric Series (1959), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · JFM 58.0296.09
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.