Summation of Fourier Series

A general summability method of different orthogonal series is given with the help of an integrable function θ. As special cases the trigonometric Fourier, Walsh-, Walsh-Kaczmarz-, Vilenkinand Ciesielski-Fourier series and the Fourier transforms are considered. For each orthonormal system a different Hardy space is introduced and the atomic decomposition of these Hardy spaces are presented. A sufficient condition is given for a sublinear operator to be bounded on the Hardy spaces. Under some conditions on θ it is proved that the maximal operator of the θ-means of these Fourier series is bounded from the Hardy space Hp to Lp (p0 < p ≤ ∞) and is of weak type (1,1), where p0 < 1 is depending on θ. In the endpoint case p = p0 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 Cesàro, Fejér, Riesz, de La Vallée-Poussin, Rogosinski, Weierstrass, Picar, Bessel and Riemann summations. Similar results are verified for several-dimensional Fourier series and Hardy spaces and for the multidimensional dyadic derivative.