Θ-summability of Fourier series

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.