Marcinkiewicz-summability of multi-dimensional Fourier transforms and Fourier series

A generalization of Marcinkiewicz-summability of multi-dimensional Fourier transforms and Fourier series is investigated with the help of a continuous function theta. Under some weak conditions on theta we show that the maximal operator of the Marcinkiewicz-theta-means of a tempered distribution is bounded from H(p) (X(d)) to L(p)(X(d)) for all d/(d + alpha) <= infinity and, consequently, is of weak type (1,1), where 0 < alpha <= 1 is depending only on theta and X=R or X = T. As a consequence we obtain a generalization of a summability result due to Marcinkiewicz and Zhizhiashvili for d-dimensional Fourier transforms and Fourier series, more exactly, the Marcinkiewicz-theta-means of a function f is an element of L(1) (X(d)) converge a.e. to f. Moreover, we prove that the Marcinkiewicz-theta-means are uniformly bounded on the spaces H(p)(X(d)) and so they converge in norm (d/(d + alpha) < p < infinity). Similar results are shown for conjugate functions. Some special cases of the Marcinkiewicz-theta-summation are considered, such as the Fejer, Cesaro, Weierstrass, Picar, Bessel, de La Vallee-Poussin, Rogosinski and Riesz summations. (C) 2011 Elsevier Inc. All rights reserved.