On the Fejér means of bounded Ciesielski systems
Tom 146 / 2001
Studia Mathematica 146 (2001), 227-243 MSC: Primary 46B15, 41A15; Secondary 42B08. DOI: 10.4064/sm146-3-2
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 $.