ON CONVERGENCE OF THE FOURIER DOUBLE SERIES WITH RESPECT TO THE VILENKIN SYSTEMS

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 \mathclose{-} \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) dx dy \ ; $ $ \lim\limits_{R \in \Gamma,\ R \to \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) $.