θ-summation and Hardy spaces

A general summability method of Fourier series and Fourier transforms is given with the help of an integrable function theta having integrable Fourier transform. Under some weak conditions on theta we show that the maximal operator of the theta -means of a distribution is bounded from H-p(T) to L-p(T) (p(0) < p < infinity) and is of weak type (1,1), where H-p(T) is the classical Hardy space and p(0) < 1 is depending only on II. As a consequence we obtain that the <theta>-means of a function f epsilon L-1(T) converge a.e. to f. For the endpoint p(0) we get that the maximal operator is of weak type (H-p theta(T), L-p0(T)). Moreover, we prove that the theta -means are uniformly bounded on the spaces H-p(T) whenever p0 < p < infinity and are uniformly of weak type (H-p0(T), H-p0(T)). Thus, in the case f epsilon H-p(T), the theta -means converge to f in H-p(T) norm (p(0) < p < infinity). The same results are proved for the conjugate theta -means and for Fourier transforms, too. Some special cases of the theta -summation are considered, such as the Weierstrass, Picar, Bessel, Fejer, Riemann, de La Vallee-Poussin; Rogosinski and Riesz summations. (C) 2000 Academic Press.