ON THE FOURIER-VILENKIN COEFFICIENTS

In this article, we prove the following statement that is true for both unbounded and bounded Vilenkin systems： for any ε∈ （0, 1）, there exists a measurable set E C [0, 1） of measure bigger than 1 - s such that for any function f ∈ LI[0, 1）, it is possible to find a function g ∈ L^1 [0, 1） coinciding with f on E and the absolute values of non zero Fourier coefficients of g with respect to the Vilenkin system are monotonically decreasing.