Simon  proved that the maximal operator of (C, α)-means of Fourier series with respect to the Walsh-Kaczmarz system is bounded from the martingale Hardy space Hp to the space Lp for p > 1/(1 + α). In this paper we prove that this boundedness result does not hold if p ≦ 1/(1 + α). However, in the endpoint case p = 1/(1 + α) the maximal operator σ*α,k is bounded from the martingale Hardy space H1/(1+α) to the space weak-L1/(1+α).
The main aim of this paper is to prove that the maximal operator σ0k*:= supn ∣σn,nk∣ of the Fej�r means of double Fourier series with respect to the Kaczmarz system is not bounded from the Hardy space H1/2 to the space weak-L1/2.
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.