theta-summation and Hardy spaces

A general summability method of Fourier series and Fourier transforms is gi ven with the help of an integrable function theta having integrable Fourier transform. Under some weak conditions on theta we show that the maximal op erator 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 cla ssical Hardy space and p(0) < 1 is depending only on II. As a consequence w e 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 t ype (H-p theta(T), L-p0(T)). Moreover, we prove that the theta -means are u niformly bounded on the spaces H-p(T) whenever p0 < p < infinity and are un iformly 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 tran sforms, too. Some special cases of the theta -summation are considered, suc h as the Weierstrass, Picar, Bessel, Fejer, Riemann, de La Vallee-Poussin; Rogosinski and Riesz summations. (C) 2000 Academic Press.