Resumen de Summability and integrability of Vilenkin series

Muharem Avdispahic, Medo Pepic

• A sufficient condition on a triangular matrix $$\Lambda = [\lambda_{nk}],(n,k\in\mathbb{N}_0), \lambda_{n0}=1\qquad (\forall n\in\mathbb{N}_0),$$ is given, in order that $$\parallel f-L_n(\Lambda,f )\parallel_q\rightarrow 0\quad (n\rightarrow\infty), $$ for $q\in\lfloor 1,\infty\rfloor$ and an arbitrary function $f\in L^q(G)$, where $$L_n(\Lambda,f):=\sum^n_{k=0}\lambda_{nk}\hat{f}(k)_{\mathcal{X}k}, (n\in\mathbb{N}_0), \textrm{ with } \hat{f}(k):=\int\limits_G f_{\bar{\mathcal{X}}k},$$ is a sequence of linear operators on $L^1(G)$ associated to the matrix $\Lambda$. This generalizes an earlier result of Blyumin from bounded to general Vilenkin groups. A new integrability class for general Vilenkin groups, larger than the class $F^\ast_p(G)$ from [3] is established.