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

Let {Wk(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)∈L1(E) there exists a function g(x,y)∈L1[0,1)2, satisfying the following conditions: g(x,y)=f(x,y) on E ; the nonzero members of the sequence {|ck,s(g)|} are monotonically decreasing in all rays, where ck,s(g)=∫01∫01g(x,y)Wk―(x)Ws―(y)dxdy ; limR∈Γ, R→∞SR((x,y),g)=g(x,y) almost everywhere on [0,1)2, where SR((x,y),g)=∑k2+s2≤R2ck,s(g)Wk(x)Ws(y).