l1-Summability of d-Dimensional Fourier Transforms

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