On summability in \(L^p\)-norm on general Vilenkin groups
Let \((\chi_k, k\in \mathbb{N})\) be the Vilenkin system, \(\widehat f(k)\) be the \(k\)th Vilenkin-Fourier coefficient and \[ L_n(\Lambda, f)= \sum^n_{k=0} \lambda_{n,k}\widehat f(k)\chi_k, \] where \(\Lambda= [\lambda_{n,k}]\) and \(\lambda_{n,0}= 1\), \(\lambda_{n,k}= 0\) for all \(k> n\) \((n\in\mathbb{N})\). Sufficient conditions are given such that \(\lim_{n\to\infty} L_n(\Lambda, f)= f\) in \(L^q\) norm \((1\leq q\leq\infty)\). If the Vilenkin system is bounded then this result coincides with an earlier one of Blyumin.
scientific article; zbMATH DE number 1512910