Marcinkiewicz-φ-summability of Fourier transforms

A general summability method of two-dimensional Fourier transforms is given with the help of an integrable function theta. Under some conditions on theta we show that the maximal operator of the Marcinkiewicz-theta-means of a tempered distribution is bounded from H(p)(R(2)) to L(p)(R(2)) for all p(o) < pless than or equal to infinity and, consequently, is of weak type (1, 1), where p(o) < 1 depends only on theta. As a consequence we obtain a generalization for Fourier transforms of a summability result due to Marcinkievicz and Zhizhiashvili, more exactly, the Marcinkiewicz-theta-means of a function f is an element of L(1)(R(2)) converge a.e. to the function in question. Moreover, we prove that the Marcinkiewicz-theta-means are uniformly bounded on the spaces H(p)(R(2)) and so they converge in norm (p(o) < p < infinity). Some special cases of the Marcinkiewicz-theta-summation are considered, such as the Weierstrass, Picar, Bessel, Fejer, de la Vallee-Poussin, Rogosinski and Riesz summations.