Norm and almost everywhere convergence of matrix transform means of Walsh-Fourier series

Abstract

Abstract We show the uniformly boundedness of the L1 norm of general matrix transform kernel functions with respect to the Walsh-Paley system. Special such matrix means are the well-known Cesàro, Riesz, Bohner-Riesz means. Under some conditions, we verify that the kernels K n T = ∑ k = 1 n t k , n D k {\rm{K}}_{\rm{n}}^{\rm{T}} = \sum\nolimits_{{\rm{k = 1}}}^{\rm{n}} {{{\rm{t}}_{{\rm{k}},{\rm{n}}}}{{\rm{D}}_{\rm{k}}}} , (where Dk is the kth Dirichlet kernel) satisfy ‖ K n T ‖ 1 ≤ c . {\left\| {{\rm{K}}_{\rm{n}}^{\rm{T}}} \right\|_1} \le {\rm{c}}{\rm{.}} As a result of this we prove that for any 1 ≤ p < ∞ and f ∈ Lp the Lp-norm convergence ∑ k = 1 n t k , n S k ( f ) → f \sum\nolimits_{{\rm{k = 1}}}^{\rm{n}} {{{\rm{t}}_{{\rm{k}},{\rm{n}}}}{{\rm{S}}_{\rm{k}}}\left( {\rm{f}} \right)} \to {\rm{f}} holds. Besides, for each integrable function f we have that these means converge to f almost everywhere.