ON CONVERGENCE OF THE FOURIER DOUBLE SERIES WITH RESPECT TO THE VILENKIN SYSTEMS - Simonyan, L.S. (Proceedings of the YSU A: Physical and Mathematical Sciences, 2018)
Let { W_k (x) } _k = 0^∞ be either unbounded or bounded Vilenkin system. Then, for each 0 < ε < 1 , there exist a measurable set E [0,1)^2 of measure |E| > 1 - ε , and a subset of natural numbers Γ of density 1 such that for any function f(x,y) L^1 (E) there exists a function g(x,y) 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) = _0^1 _0^1 g(x, y) W_k(x) W_s(y) dx dy ; _R Γ, R → ∞ S_R((x,y),g) = g(x,y) almost everywhere on [0,1)^2 , where S_R((x,y),g) = _k^2+s^2 ≤ R^2 c_k, s(g) W_k(x) W_s(y) .