On the Fejér means of bounded Ciesielski systems
We investigate the bounded Ciesielski systems, which can be obtained from the spline systems of order $(m,k)$ in the same way as the Walsh system arises from the Haar system. It is shown that the maximal operator of the Fejér means of the Ciesielski–Fourier series is bounded from the Hardy space $H_p$ to $L_p$ if $1/2< p< \infty $ and $m\geq 0$, $|k|\leq m+1$. Moreover, it is of weak type $(1,1)$. As a consequence, the Fejér means of the Ciesielski–Fourier series of a function $f$ converges to $f$ a.e. if $f \in L_1$ as $n\to \infty $.