Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres
DOI:
https://doi.org/10.37863/umzh.v73i3.196Keywords:
Cesàro means with varying parameters, two-dimensional Walsh-Fourier series, Marcinkiewicz meansAbstract
UDC 517.5
We prove that the maximal operator of some (C,βn) means of cubical partial sums of two variable Walsh – Fourier series of integrable functions is of weak type (L1,L1). Moreover, the (C,βn)-means σβn2nf of the function f∈L1 converge a.e. to f for f∈L1(I2), where I is the Walsh group for some sequences 1>βn↘0.
References
T. Akhobadze, On the convergence of generalized Ces`aro means of trigonometric Fourier series. I, Acta Math. Hungar. 115, № 1-2, 59 – 78 (2007), https://doi.org/10.1007/s10474-007-5214-7 DOI: https://doi.org/10.1007/s10474-007-5214-7
T. Akhobadze, On the generalized Ces`aro means of trigonometric Fourier series, Bull. TICMI, 18, № 1, 75 – 84 (2014).
A. Abu Joudeh, G. G´at, Almost everywhere convergence of Ces`aro means with varying parameters of Walsh – Fourier series, Miskolc Math. Notes, 19, № 1, 303 – 317 (2018).
M. I. D’yachenko, On (C,alpha)-summability of multiple trigonometric Fourier series, (Russian) Soobshch. Akad. Nauk Gruzin. SSR 131, № 2, 261 – 263 (1988).
G. G´at, Convergence of Marcinkiewicz means of integrable functions with respect to two-dimensional Vilenkin systems, Georgian Mathematical Journal. 11, № 3, 467 – 478 (2004).
G. G´at, On (C,1) summability for Vilenkin-like systems, Stud. Math. 144, № 2, 101 – 120 (2001),https://doi.org/10.4064/sm144-2-1 DOI: https://doi.org/10.4064/sm144-2-1
U. Goginava, Marcinkiewicz-Fejer means of d-dimensional Walsh – Fourier series, Journal of Mathematical Analysis and Applications. 307, № 1, 206 – 218 (2005), https://doi.org/10.1016/j.jmaa.2004.11.001 DOI: https://doi.org/10.1016/j.jmaa.2004.11.001
U. Goginava, Almost everywhere convergence of (C,alpha)-means of cubical partial sums of d-dimensional Walsh – Fourier series, Journal of Approximation Theory. 141, № 1, 8 – 28 (2006), https://doi.org/10.1016/j.jat.2006.01.001 DOI: https://doi.org/10.1016/j.jat.2006.01.001
J. Marcinkiewicz, Sur une nouvelle condition pour la convergence presque partout des s´eries de Fourier (French), Annali della Scuola Normale Superiore di Pisa-Classe di Scienze. 8, № 3-4, 239 – 240 (1939), https://doi.org/10.4064/sm-8-1-78-91 DOI: https://doi.org/10.4064/sm-8-1-78-91
F. Schipp, W.R. Wade, P. Simon, J. P´al, Walsh series: an introduction to dyadic harmonic analysis, Adam Hilger, Bristol, New York (1990).
F. Weisz, Convergence of double Walsh – Fourier series and Hardy spaces., Approxim. Theory and Appl., 17, № 2, 32 – 44 (2001), https://doi.org/10.1023/A:1015553812707 DOI: https://doi.org/10.1023/A:1015553812707
L. V. Žižiašvili, A generalization of a theorem of Marcinkiewicz., Izv. Ross. Akad. Nauk. Ser. Mat., 32, № 5, 1112 – 1122 (1968).
A. Zygmund, Trigonometric series, Univ. Press, Cambridge (1959).
