On the Fejér means of bounded Ciesielski systems

Ferenc Weisz¹

¹ Department of Numerical Analysis Eötvös L. University Pázmány P. sétány 1/D H-1117 Budapest, Hungary

Studia Mathematica, Tome 146 (2001) no. 3, pp. 227-243

Résumé

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 $.

Détail

DOI : 10.4064/sm146-3-2

Keywords: investigate bounded ciesielski systems which obtained spline systems order walsh system arises haar system shown maximal operator fej means ciesielski fourier series bounded hardy space infty geq leq moreover weak type consequence fej means ciesielski fourier series function converges infty