Some new resuts concering strong convergence of Fejér means with respect to Vilenkin systems

UDC 517.5 We prove some new strong convergence theorems for partial sums and Fej\'er means with respect to the Vilenkin system.

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 \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) $.

On the Coefficients of Multiple Series with Respect to Vilenkin System

We give a sufficient condition for coefficients of double series Σ Σn,m an,m χn,m with respect to Vilenkin system to be convergent to zero when n + m → ∞. This result can be applied to the problem of recovering coefficients of a Vilenkin series from its sum.