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

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