On some subsequences of Fejér means for integrable functions on unbounded Vilenkin groups
Let \(\{m_0, m_1, \dots\}\) be an unbounded sequence of integers not less than 2. We denote by \(P\) the set of positive integers and let \(\mathbb N=P\cup \{0\}\). Let \(G:=\prod^\infty_{n=0}\mathbb Z(m_n)\), where \(Z(m_n)\) denotes the discrete group of order \(m_n\), with addition \(\pmod{m_n}\). Each element \(x\) from \(G\) can be represented as a sequence \((x_n)\), where \(x_n\in \{0,1,\dots, m_n-1\}\) for every integer \(n\geq 0\). Addition in \(G\) is obtained coordinatewise. Define the sequence \((M_n)^\infty_{n=0}\) as follows: \(M_0=1\) and \(M_{n+1}=m_nM_n\). The generalized Rademacher functions are defined by \(r_n(x)=\exp(2\pi ix_n/m_n)\), \(n\in\mathbb N\), \(x\in G\). The Vilenkin system generated by \(\{m_0, m_1, \dots\}\) is given by \(\psi_n(x)=\prod^\infty_{i=0}r^{n_1}_i(x)\), \(x\in G\), where \(n\in\mathbb N\) is written in the form \(n=\sum^\infty_{i=0}n_iM_i\), \(n_i\in \mathbb N\cap [0,m_i)\). The Fourier coefficients and partial sums of the Fourier series are respectively defined as follows \[ \widehat{f}(n)=\int_{G}f(x)\overline{\psi_n(x)}\,dx, \quad n\in\mathbb N; \quad S_n(f)(x)=\sum^{n-1}_{k=0}\widehat{f}(k)\psi_k(x), \quad n\in P. \] The main result of paper is Theorem 1. Let \(f\in L^1(G)\), \(L,S\in P\) be fixed. Then the partial sums \(S_{a_nM_n+\dots +a_{n+l}M_{n+l}}f\) converge to \(f\) almost everywhere uniformly in \(1\leq l\leq L\), \(a_{N+t}\in \{0,1, \dots,\min(S,m_{N+t}-1)\}\) for \(0\leq t\leq l\), where, in addition, \(a_N,a_{N+l}\neq 0\).
scientific article; zbMATH DE number 6810254