Geodesic
On convergence of the Fourier double series with respect to the Vilenkin systems
Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 52 (2018) no. 1, pp. 12-18.

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Let $\{W_{k}(x)\}_{k=0}^{\infty}$ be either unbounded or bounded Vilenkin system. Then, for each $0\varepsilon1$, there exist a measurable set $E\subset[0,1)^{2}$ of measure $|E|>1-\varepsilon$, and a subset of natural numbers $\Gamma$ of density $1$ such that for any function $f(x,y)\in L^{1}(E)$ there exists a function $g(x,y)\in L^{1}[0,1)^{2}$, satisfying the following conditions: $g(x,y)=f(x,y)$ on $E$; the nonzero members of the sequence $\{|c_{k,s}(g)|\}$ are monotonically decreasing in all rays, where $c_{k,s}(g)=\int\limits_{0}^{1}\int\limits_{0}^{1}g(x,y)\overline{{W}_{k}}(x)\overline{W_{s}}(y)dxdy$; $\displaystyle\lim_{R\in \Gamma,\ R\rightarrow\infty}S_{R}((x,y),g)=g(x,y)$ almost everywhere on $[0,1)^2$, where $S_{R}((x,y),g)=\sum\limits_{k^{2}+s^{2}\leq R^{2}}c_{k,s}(g)W_{k}(x)W_{s}(y)$.
Keywords: Vilenkin system, convergence almost everywhere
Mots-clés : Fourier coefficients. 
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L. S. Simonyan. On convergence of the Fourier double series with respect to the Vilenkin systems. Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 52 (2018) no. 1, pp. 12-18. http://geodesic.mathdoc.fr/item/UZERU_2018_52_1_a2/

[1] Kluwer Academic Publishers, Dordrecht, 1991 | MR | MR

[2] G. N. Agaev, N. Ja. Vilenkin, G. M. Dzhafarli, A. I. Rubinshtejn, Multiplicative Systems of Functions and Harmonic Analysis on Zero-Dimensional Groups, Ehlm, Baku, 1981, 180 pp. (in Russian) | MR

[3] N. Ja. Vilenkin, “On Class of Complete Orthonormal Systems”, Amer. Math. Soc. Transl., 28:2 (1963), 1–35 | MR

[4] S. L. Blyumin, “Certain Properties of a Class of Multiplicative Systems and Questions of Approximation of Functions by Polynomials in These Systems”, Izv. Vyssh. Uchebn. Zaved., 1968, no. 4, 13–22 (in Russian) | MR

[5] J. A. Gosselin, “Convergence a. e. of Vilenkin–Fourier Series”, Trans. Amer. Math. Soc., 185 (1973), 345–370 | DOI | MR

[6] J. J. Price, “Certain Groups of Orthonormalstep Functions”, Canad. J. Math., 93 (1957), 413–425 | DOI | MR

[7] A. M. Zubakin, “The Correction Theorems of Men'shov for a Certain Class of Multiplicative Orthonormal Systems of Functions”, Izv. Vyssh. Uchebn. Zaved., 1969, no. 12, 34–46 (in Russian) | MR

[8] C. Watari, “On Generalized Walsh–Furier Series”, Tohoku Math. J., 73:8 (1958), 435–438 | MR

[9] W. S. Young, “Mean Convergence of Generalized Walsh–Fourier Series”, Trans. Amer. Math. Soc., 218 (1976), 311–320 | DOI | MR

[10] P. Billard, “Sur la Convergence Presque Partout des Series de Furier–Walsh des Fonctions de l'Espace $L^2(0,1)$”, Studia Math., 28:3 (1967), 363–388 | DOI | MR

[11] M. G. Grigorian, “On Some Properties of Orthogonal Systems”, Izv. Math. Russian Acad. Sci., 43:2 (1994) | MR

[12] M. G. Grigorian, “On the $L_{\mu}^{p}$-Strong Property of Orthonormal Systems”, Math. Sbornik, 194:10 (2003), 1503–1532 | DOI | MR

[13] M. G. Grigorian, “Modifications of Functions, Fourier Coefficients and Nonlinear Approximation”, Math. Sbornik, 203:3 (2012), 351–379 | DOI | MR

[14] M. G. Grigorian, S. A. Sargsyan, “On the Fourier–Vilenkin Coefficients”, Acta Mathematica Scientia B, 37:2 (2017), 293–300 | DOI | MR

[15] M. G. Grigorian, “On Convergence of Fourier Series in Complete Orthonormal Systems in the $L^{1}$ Metric and Almost Everywhere”, Math. USSR-Sb, 70:2 (1991), 445–466 | DOI | MR