Some new resuts concering strong convergence of Fejér means with respect to Vilenkin systems

L.-E.  Persson; G.  Tephnadze; G. Tutberidze; P.  Wall

doi:10.37863/umzh.v73i4.226

L.-E. Persson UiT -The Arctic Univ. Narvik, Norway and Karlstad Univ., Sweden
G. Tephnadze Univ. Georgia, School Sci. and Technology, Tbilisi, Georgia
G. Tutberidze Univ. Georgia, School Sci. and Technology, Georgia and UiT - The Arctic Univ. Narvik, Norway
P. Wall Lule˚a Univ. Technology, Sweden

DOI: https://doi.org/10.37863/umzh.v73i4.226

Ключові слова: Vilenkin system, Fejér means, martingale Hardy space, strong convergence

Анотація

УДК 517.5

Деякi новi результати щодо строгої збiжностi середнiх Феєра вiдносно систем Вiленкiна

Доведено деякі теореми про строгу збіжність часткових сум та середніх Феєра відносно системи Віленкіна.

Посилання

G. N. Agaev, N. Ya. Vilenkin, G. M. Dzhafarly, A. I. Rubinshtein, Multiplicative systems of functions and harmonic analysis on zero-dimensional groups, Elm, Baku (1981) (in Russian).

I. Blahota, G. G`at, U. Goginava, Maximal operators of Fej´er means of double Vilenkin – Fourier series, Colloq. Math., 107, № 2, 287 – 296 (2007), https://doi.org/10.4064/cm107-2-8 DOI: https://doi.org/10.4064/cm107-2-8

I. Blahota, G. G`at, U. Goginava, Maximal operators of Fej´er means of Vilenkin – Fourier series, J. Inequal. Pure and Appl. Math., 7, 1 – 7 (2006).

I. Blahota, G. Tephnadze, Strong convergence theorem for Vilenkin – Fej´er means, Publ. Math. Debrecen, 85, № 1 – 2, 181 – 196 (2014), https://doi.org/10.5486/PMD.2014.5896 DOI: https://doi.org/10.5486/PMD.2014.5896

G. G`at, Inverstigations of certain operators with respect to the Vilenkin system, Acta Math. Hung., 61, 131 – 149 (1993), https://doi.org/10.1007/BF01872107 DOI: https://doi.org/10.1007/BF01872107

U. Goginava, L. D. Gogoladze, Strong convergence of cubic partial sums of two-dimensional Walsh – Fourier series, Constructive Theory of Functions (Sozopol, 2010), In memory of Borislav Bojanov, Acad. Publ., House, Sofia (2012), p. 108 – 117.

L. D. Gogoladze, On the strong summability of Fourier series, Bull. Acad. Sci. Georgian SSR, 52, № 2, 287 – 292 (1968).

B. I. Golubov, A. V. Efimov, V. A. Skvortsov, Walsh series and transforms, Nauka, Moscow (1987) (in Russian).

N. Memi´c, I. Simon, G. Tephnadze, Strong convergence of two-dimensional Vilenkin – Fourier series, Math. Nachr., 289, № 4, 485 – 500 (2016), https://doi.org/10.1002/mana.201400239 DOI: https://doi.org/10.1002/mana.201400239

F. Schipp, W. R. Wade, P. Simon, J. P´al, Walsh series. An introduction to dyadic harmonic analysis, Adam Hilger, Ltd., Bristol (1990).

P. Simon, Strong convergence of certain means with respect to the Walsh – Fourier series, Acta Math. Hung., 49, № 1 – 2, 425 – 431 (1987), https://doi.org/10.1007/BF01951006 DOI: https://doi.org/10.1007/BF01951006

P. Simon, Strong convergence theorem for Vilenkin – Fourier series, J. Math. Anal. and Appl., 245, 52 – 68 (2000), https://doi.org/10.1006/jmaa.2000.6732 DOI: https://doi.org/10.1006/jmaa.2000.6732

L.-E. Persson, G. Tephnadze, P.Wall, Some new (Hp,Lp) type inequalities of maximal operators of Vilenkin – N¨orlund means with non-decreasing coefficients, J. Math. Inequal., 9, № 4, 1055 – 1069 (2015), https://doi.org/10.7153/jmi-09-82 DOI: https://doi.org/10.7153/jmi-09-82

G. Tephnadze, Fej´er means of Vilenkin – Fourier series, Stud. Sci. Math. Hung., 49, № 1, 79 – 90 (2012), https://doi.org/10.1556/SScMath.2011.1187 DOI: https://doi.org/10.1556/sscmath.2011.1187

G. Tephnadze, Martingale Hardy spaces and summability of the one dimensional Vilenkin – Fourier series, PhD thesis, Lule˚a Univ. Technology (2015).

G. Tephnadze, A note on the Fourier coefficients and partial sums of Vilenkin – Fourier series, Acta Math. Acad. Paedagog. Nyh´azi., 28, 167 – 176 (2012).

G. Tephnadze, Strong convergence theorems of Walsh – Fej´er means, Acta Math. Hungar., 142, № 1, 244 – 259 (2014), https://doi.org/10.1007/s10474-013-0361-5 DOI: https://doi.org/10.1007/s10474-013-0361-5

G. Tephnadze, Strong convergence of two-dimensional Walsh – Fourier series, Ukr. Math. J., 65, № 6, 822 – 834 (2013), https://doi.org/10.1007/s11253-013-0828-0 DOI: https://doi.org/10.1007/s11253-013-0828-0

G. Tutberidze, A note on the strong convergence of partial sums with respect to Vilenkin system, J. Contemp. Math. Anal., 54, № 6, 319 – 324 (2019).

N. Ya. Vilenkin, On a class of complete orthonormal systems, Izv. Akad. Nauk USSR, Ser. Mat., 11, 363 – 400 (1947).

F. Weisz, Martingale Hardy spaces and their applications in Fourier analysis, Springer, Berlin etc. (1994), https://doi.org/10.1007/BFb0073448 DOI: https://doi.org/10.1007/BFb0073448

F. Weisz, Hardy spaces and Ces`aro means of two-dimensional Fourier series, Bolyai Soc. Math. Stud., 353 – 367 (1996).

F. Weisz, Strong convergence theorems for two-parameter Walsh – Fourier and trigonometric-Fourier series, Stud. Math., 117, № 2, 173 – 194 (1996), https://doi.org/10.4064/sm-117-2-173-194 DOI: https://doi.org/10.4064/sm-117-2-173-194

F. Weisz, Ces`aro summability of one and two-dimensional Fourier series, Anal. Math., 5, 353 – 367 (1996).