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

L. S. Simonyan

Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 52 (2018) no. 1, pp. 12-18

Résumé

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 $S_{R}((x,y),g)=\sum\limits_{k^{2}+s^{2}\leq R^{2}}c_{k,s}(g)W_{k}(x)W_{s}(y)$.

Détail

Keywords: Vilenkin system, convergence almost everywhere
Mots-clés : Fourier coefficients.