On divergence of Fejér means with respect to Walsh system on sets of measure zero (Q6611668)

Let \(\mathbb{N}_+\) denote the set of the positive integers and \(\mathbb{N}=\mathbb{N}_+\cup\{0\}\). Let us consider \(\mathbb{Z}_2=\{0,1\}\) and the group \(G\) as the complete direct product group \(\mathbb{Z}_2\). It is known that \(n=\sum_{j=0}^{\infty}n_j2^j\), for every \(n\in\mathbb{N}\), where \(n_j\in \mathbb{Z}_2\). Then, the generalized Rademacher functions are defined as,\N\[\Nr_k(x)=(-1)^{x_k},\N\]\Nwith \(x\in G\) and \(k\in \mathbb{N}\). Also, the Walsh system \((\omega_n)_n\) is defined as,\N\[\N\omega_n(x)=\prod_{k=0}^{\infty}r_k^{n_k}(x), \quad n\in \mathbb{N}.\N\]\NIn this context, Fourier coefficients, partial sums of the Fourier series, and Dirichlet kernels are defined by,\N\[\N\hat{f}(n)=\int_G f\omega_n d\mu, \quad (n\in\mathbb{N})\N\]\N\[\NS_nf=\sum_{k=0}^{n-1}\hat{f}(k)w_k \quad \text{and} \quad D_n=\sum_{k=0}^{n-1}\omega_k, \quad (n\in\mathbb{N}_+),\N\]\Nrespectively, where \(\mu\) is the Haar measure on \(G\). Then, for any \(n\in\mathbb{N}_+\), the Féjer means are defined by\N\[\N\sigma_nf=\frac{1}{n}\sum_{k=0}^{n-1}S_kf \quad \text{and} \quad K_n=\frac{1}{n}\sum_{k=0}^{n-1}D_k\N\]\Nis the Féjer kernel.\N\NThe main result obtained by the authors is the following theorem.\N\NTheorem 3.1. Let \(1\leq p<\infty\). If \(E\subseteq G\) is a set of measure zero, then there exists a function in \(L_p(G)\), such that Féjer means with respect to the Walsh system diverge on this set.\N\NIn the words of the authors themselves: ``In fact this Theorem 3.1 follows from the general result of \textit{G. A. Karagulyan} [Sb. Math. 202, No. 1, 9--33 (2011; Zbl 1217.42009); translation from Mat. Sb. 202, No. 1, 11--36 (2011)], but we provide an alternative approach and the constructed function has a simple and explicit representation''.\N\NFor the entire collection see [Zbl 1544.35009].