A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions
on Θ we show that if the maximal Fejér operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin
and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means
of these Fourier series is bounded from Hp to Lp (1/2<p≤; ∞) and is of weak type (1,1). In the endpoint case p=1/2 a weak type inequality is derived. As a consequence we obtain that the Θ-means of a function f∈L1 converge a.e. to f. Some special cases of the Θ-summation are considered, such as the Weierstrass, Picar, Bessel, Riesz, de la Vallée-Poussin,
Rogosinski and Riemann summations. Similar results are verified for several-dimensional Fourier series and Hardy spaces.
We investigate the Kronecker product of bounded Ciesielski systems, which can be obtained from the spline systems of order
(m, k) in the same way as the Walsh system from the Haar system. It is shown that the maximal operator of the Fejér means of the
d-dimensional Ciesielski-Fourier series is bounded from the Hardy space Hp([0, 1)d1×…×[0, 1)dl to Lp ([0, 1)d) if 1/2<p<∞ and mj≥0, ‖kj‖≤mj+1. By an interpolation theorem, we get that the maximal operator is also of weak type (