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Cesáro and Riesz summability with varying parameters of multi-dimensional Walsh-Fourier series. (English) Zbl 1463.42070

Firstly (Theorem 2) the author gives a simpler proof of his estimate for the Cesàro kernels \( K_n^{(\alpha)} \) of the Walsh-Fourier series [Anal. Math. 27, 141–155 (2001; Zbl 0992.42016)], Theorem 1, an estimate of the Riesz kernels \( K_n^{(\alpha,\gamma)} \) and the dependence of the constants of the parameters \( \alpha, \gamma \). Then modifications of the Cesàro and Riesz means are introduced by replacing the corresponding kernels by their estimates from Theorem 2. The main result gives an estimate for the related maximal operator with varying parameters. This result generalizes the main result of Gy. Gát and U. Goginava [Acta Math. Hung. 159, No. 2, 653–668 (2019; Zbl 1449.42045)]. Actually the results are proved for the more general case \( n\in \mathbb{N}_{\alpha,v} \) (here the almost everywhere convergence of the Cesàro means was proved in A. A. A. Joudeh and G. Gát [Miskolc Math. Notes 19, No. 1, 303–317 (2018; Zbl 1463.42066)]). Analogous results are proved for multi-dimensional functions as well.

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42A24 Summability and absolute summability of Fourier and trigonometric series
42B25 Maximal functions, Littlewood-Paley theory
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[1] T. Akhobadze, On the convergence of generalized Cesàro means of trigonometric Fourier series. I, Acta Math. Hungar., 115 (2007), 59-78 · Zbl 1136.42004
[2] P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Birkhäuser (Basel, 1971) · Zbl 0217.42603
[3] Chang, SYA; Fefferman, R., Some recent developments in Fourier analysis and \(H^p\)-theory on product domains, Bull. Amer. Math. Soc., 12, 1-43 (1985) · Zbl 0557.42007 · doi:10.1090/S0273-0979-1985-15291-7
[4] Coifman, RR; Weiss, G., Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83, 569-645 (1977) · Zbl 0358.30023 · doi:10.1090/S0002-9904-1977-14325-5
[5] Fefferman, R., Calderon-Zygmund theory for product domains: \(H^p\) spaces, Proc. Nat. Acad. Sci. USA, 83, 840-843 (1986) · Zbl 0602.42023 · doi:10.1073/pnas.83.4.840
[6] Fejér, L., Untersuchungen über Fouriersche Reihen, Math. Ann., 58, 51-69 (1904) · JFM 34.0287.01 · doi:10.1007/BF01447779
[7] Fine, NJ, Cesàro summability of Walsh-Fourier series, Proc. Nat. Acad. Sci. USA, 41, 558-591 (1955) · Zbl 0065.05303 · doi:10.1073/pnas.41.8.588
[8] Fujii, N., A maximal inequality for \({H}^1\)-functions on a generalized Walsh-Paley group, Proc. Amer. Math. Soc., 77, 111-116 (1979) · Zbl 0415.43014
[9] Gát, G.; Goginava, U., Maximal operators of Cesàro means with varying parameters of Walsh-Fourier series, Acta Math. Hungar., 159, 653-668 (2019) · Zbl 1449.42045 · doi:10.1007/s10474-019-00961-2
[10] Goginava, U., The maximal operator of the Marcinkiewicz-Fejér means of \(d\)-dimensional Walsh-Fourier series, East J. Approx., 12, 295-302 (2006) · Zbl 1487.42067
[11] L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education (New Jersey, 2004) · Zbl 1148.42001
[12] Herz, C., Bounded mean oscillation and regulated martingales, Trans. Amer. Math. Soc., 193, 199-215 (1974) · Zbl 0321.60041 · doi:10.1090/S0002-9947-1974-0353447-5
[13] Joudeh, AAA; Gát, G., Convergence of Cesàro means with varying parameters of Walsh-Fourier series, Miskolc Math. Notes, 19, 303-317 (2018) · Zbl 1463.42066 · doi:10.18514/MMN.2018.2347
[14] Lebesgue, H., Recherches sur la convergence des séries de Fourier, Math. Ann., 61, 251-280 (1905) · JFM 36.0330.01 · doi:10.1007/BF01457565
[15] Móricz, F.; Schipp, F.; Wade, WR, Cesàro summability of double Walsh-Fourier series, Trans. Amer. Math. Soc., 329, 131-140 (1992) · Zbl 0795.42016
[16] Riesz, M., Sur la sommation des séries de Fourier, Acta Sci. Math. (Szeged), 1, 104-113 (1923) · JFM 49.0205.01
[17] Schipp, F., Über gewissen Maximaloperatoren, Ann. Univ. Sci. Budapest Sect. Math., 18, 189-195 (1975) · Zbl 0351.42012
[18] F. Schipp and P. Simon, On some \(({H},{L}_1)\)-type maximal inequalities with respect to the Walsh-Paley system, in: Functions, Series, Operators, Proc. Conf. in Budapest, 1980, Coll. Math. Soc. J. Bolyai, vol. 35, North Holland (Amsterdam, 1981), pp. 1039-1045 · Zbl 0535.42019
[19] Schipp, F.; Wade, WR; Simon, P.; Pál, J., Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger (1990), New York: Bristol, New York · Zbl 0727.42017
[20] Simon, P., Cesàro summability with respect to two-parameter Walsh systems, Monatsh. Math., 131, 321-334 (2000) · Zbl 0976.42014 · doi:10.1007/s006050070004
[21] Simon, P.; Weisz, F., Weak inequalities for Cesàro and Riesz summability of Walsh-Fourier series, J. Approx. Theory, 151, 1-19 (2008) · Zbl 1143.42032 · doi:10.1016/j.jat.2007.05.004
[22] Stein, EM; Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces (1971), Press (Princeton, N.J.: Princeton Univ, Press (Princeton, N.J. · Zbl 0232.42007
[23] Trigub, RM; Belinsky, ES, Fourier Analysis and Approximation of Functions, Kluwer Academic Publishers (2004), Dordrecht: Boston, London, Dordrecht · Zbl 1063.42001
[24] F. Weisz, Martingale Hardy Spaces and their Applications in Fourier Analysis, Lecture Notes in Math., vol. 1568, Springer (Berlin, 1994) · Zbl 0796.60049
[25] Weisz, F., Cesàro summability of two-parameter Walsh-Fourier series, J. Approx. Theory, 88, 168-192 (1997) · Zbl 0873.42017 · doi:10.1006/jath.1996.3021
[26] Weisz, F., \((C,\alpha )\) summability of Walsh-Fourier series, Anal. Math., 27, 141-155 (2001) · Zbl 0992.42016 · doi:10.1023/A:1014364010470
[27] Weisz, F., Summability of Multi-dimensional Fourier Series and Hardy Spaces, Mathematics and Its Applications, Kluwer Academic Publishers (2002), Dordrecht: Boston, London, Dordrecht · Zbl 1306.42003
[28] Weisz, F., Summability of multi-dimensional trigonometric Fourier series, Surv. Approx. Theory, 7, 1-179 (2012) · Zbl 1285.42010
[29] Weisz, F., Convergence and Summability of Fourier Transforms and Hardy Spaces (2017), Springer, Birkhäuser (Basel: Applied and Numerical Harmonic Analysis, Springer, Birkhäuser (Basel · Zbl 1391.42001
[30] A. Zygmund, Trigonometric Series (3rd ed.), Cambridge Press (London, 2002) · Zbl 1084.42003
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