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

We show the uniformly boundedness of theL1norm of general matrix transform kernel functions with respect to the Walsh-Paley system. Special such matrix means are the well-known Cesaro, Riesz,Bohner-Riesz means. Under some conditions, we verify that the kernels KTn=Pnk=1tk,nDk, (where Dk is the kth Dirichlet kernel) satisfy ||K-n|(T)|(1 )<= c. As a result of this we prove that for any1 <= p <infinity and f is an element of Lp theLp-norm convergence Pnk=1tk,nSk(f)-> fholds. Besides, for each integrable function f we have that these means converge to f almost everywhere