Almost Everywhere Convergence of Greedy Algorithm with Respect to Vilenkin System

Abstract

In this paper, we prove that for any ε ∈ (0, 1) there exists ameasurable set E ∈ [0, 1) with measure |E| > 1 − ε such that for any function f ∈ L¹[0, 1), it is possible to construct a function \(\tilde f \in {L^1}[0,1]\) coinciding with f on E and satisfying \(\int_0^1 {|\tilde f(x) - f(x)|dx < \varepsilon } \), such that both the Fourier series and the greedy algorithm of \(\tilde f\) with respect to a bounded Vilenkin system are almost everywhere convergent on [0, 1).