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 Proceedings of the YSU, Physics & Mathematics, 2018, Volume 52, Issue 1, Pages 12–18 (Mi uzeru452)

Mathematics

On convergence of the Fourier double series with respect to the Vilenkin systems

L. S. Simonyan

Abstract: Let $\{W_{k}(x)\}_{k=0}^{\infty}$ be either unbounded or bounded Vilenkin system. Then, for each $0<\varepsilon<1$, 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, Fourier coefficients.

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Document Type: Article
MSC: 42C20
Language: English

Citation: L. S. Simonyan, “On convergence of the Fourier double series with respect to the Vilenkin systems”, Proceedings of the YSU, Physics & Mathematics, 52:1 (2018), 12–18

Citation in format AMSBIB
\Bibitem{Sim18} \by L.~S.~Simonyan \paper On convergence of the Fourier double series with respect to the Vilenkin systems \jour Proceedings of the YSU, Physics {\&} Mathematics \yr 2018 \vol 52 \issue 1 \pages 12--18 \mathnet{http://mi.mathnet.ru/uzeru452}