Almost everywhere convergence of Fejér means of some subsequences of Fourier series for integrable functions with respect to the Kaczmarz system (Q266721)

Let \(\mathbb Z_2=\{0,1\}\) be the cyclic discrete group with addition modulo 2 and measure \(|\cdot|\) such that \(|\{0\}|=|\{0\}|=1/2\). Then \(G=\prod^\infty_{n=0}\mathbb Z_2\) is the dyadic group with product topology and measure. The elements of \(G\) are \(x=(x_n)^\infty_{n=0}\) and the \(k\)-th Rademacher function is defined by \(r_k(x)=(-1)^{x_k}\). If \(n\in\mathbb Z_+\) is written in the form \(n=\sum^\infty_{k=0}n_k2^k\), then NEWLINE\[NEWLINE\kappa_n(x)=r_{|n|}(x)\prod^{n-1}_{k=0}(r_{|n|-1-k}(x))^{n_k}, \quad n\geq 1, \quad |n|=\max\{k: n_k\neq 0\},NEWLINE\]NEWLINE and \(\kappa_0(x)=1\). This system is called Walsh-Kaczmarz system. The sum \(D^\kappa_n(x)=\sum^{n-1}_{i=0}\kappa_i(x)\) is called Walsh-Kaczmarz-Dirichlet kernel and the dyadic convolution we denote by \(*\). The main result of the paper isNEWLINENEWLINETheorem 2.1. Let \(f\in L^1(G)\) and \(\{k_n\}^\infty_{n=1}\) be a fixed sequence of positive integers. If \(\alpha(n)\in\mathbb N\) and \(\alpha(n+1)=2^{k_n}\alpha_n\), \(n\geq n_0\), then \(N^{-1}\sum^N_{n=1}D^\kappa_{\alpha(n)}*f\) converges to \(f\) a.e. on \(G\).