On Divergence of Fejér Means with Respect to Walsh System on Sets of Measure Zero

In this paper we prove by the concrete construction that for any set E of measure zero there exists a function \(f\in L_p(G) (1\leq p<\infty )\) such that the Féjer means with respect to Wals system diverge on this set. The key is to use new constructions of Walsh polynomials, which was introduced in Persson et al. (Martingale Hardy Spaces and Summability of One-dimensional Vilenkin-Fourier Series. Birkhäuser/Springer, 2022). In fact, the theorem we prove follows from the general result of Karagulyan (Mat. Sb. 202(1):11–36, 2011), but we provide an alternative approach and the constructed function in our proof has a simpler explicit representation.