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

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 KnT=∑k=1ntk,nDk{\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 ‖KnT‖1≤c.{\left\| {{\rm{K}}_{\rm{n}}^{\rm{T}}} \right\|_1} \le {\rm{c}}{\rm{.}}

Keywords

• character system
• fourierseries
• walsh-paley system
• kernel function
• matrix transform
• almost everywhere convergence
• 42c10