On the divergence of F\'ejer means with respect to Vilenkin systems on the set of measure zero

Abstract

The famous Carleson-Hunt theorem has been in focus of interest for a long time. This theorem concerns convergence almost everywhere of Fourier series of $f\in L_p$ functions for $1<p\leq \infty.$ Kolmogorov constructed a function $f\in L_1$ such that the partial sums of Fourier series diverge everywhere. On the other hand, we have boundedness result for F\'ejer means for all $1\leq p\leq \infty$. Similar results are proved for the partial sums and F\'ejer means of Vilenkin-Fourier series. But also here it appears the questions what happens on any subset $E$ of measure zero, can we even have a function which diverge there? We contribute with a new result concerning this question and prove by the concrete construction that for any set $E$ of measure zero there exists a function $f\in L_p(G_m) (1\leq p<\infty)$ such that the F\'ejer means with respect to Vilenkin systems diverge on this set, which follows similar result for the partial sums. The key is to use new constructions of Vilenkin polynomials, which was introduced in \cite{PTW2}. In fact, the theorem we prove follows from the general result of \cite{Kar}, but we provide an alternative approach and the constructed function in our proof has a simple explicit representation.