On almost everywhere convergence of the generalized Marcienkiwicz means with respect to two dimensional Vilenkin-like systems

[摘要] In this paper we investigate the almost everywhere convergence of two di\-men\-si\-onal Marcienkiwicz-like means of two variable integrable functions which is given by \[ t_{n}^{\alpha}f=\frac{1}{n}\sum_{k=0}^{{n-1}}S_{\alpha_1{(|n|, \, k)}, \alpha_2{(|n|, \, k)}}f \] $(M_{|n|} \le n < M_{|n|+1})$ and give a sufficient condition for functions $\alpha: N^2\mapsto N^2$ in order to have the a.e. relation $t_{n}^{\alpha}f \rightarrow f$ for all $ f\in L^{1}(G_{m}^2) $ with respect to two dimensional bounded Vilenkin-like systems. Finally, we give an application of the main result with respect to triangular summability of Vilenkin-like-Fourier series.

[学科分类] 数学（综合）

[关键词] almost everywhere convergence;generalized Marcienkiwicz means and two dimensional Vilenkin-like system