Abstract. A general summability method, the so-called $\theta $-summability is considered for multi-dimensional Fourier series and Fourier transforms. Under some conditions on $\theta $ we will show that the restricted maximal operator of the $\theta $-means of a distribution is bounded from $H_p({\msbm T}^d)$ to $L_p({\msbm T}^d)$ for all $p_0< p\leq\infty $ and it is of weak type $(1,1)$, provided that the supremum in the maximal operator is taken over a cone-like set. The parameter $p_0< 1$ is depending on the dimension, the function $\theta $ and on the cone-like set. As a consequence we obtain a generalization of a well-known result due to Marcinkiewicz and Zygmund, namely, that the $d$-dimensional $\theta $-means of a function $f \in L_1({\msbm T}^d)$ converge a.e. to $f$ over the cone-like set. The same results are given for Fourier transforms, too. Some special cases of the $\theta $-summation are considered, such as the Cesàro, Fejér, Riesz, Riemann, Weierstrass, Picar, Bessel, Rogosinski and de La Vallée-Poussin summations.

AMS Subject Classification (1991): 42B08, 42A38, 42A24; 42B30

Keyword(s): Hardy spaces, p, -atom, Wiener algebra, \theta, -summation of Fourier series and Fourier transforms, restricted convergence, cone-like sets

Received March 14, 2008, and in revised form June 6, 2008. (Registered under 6069/2009.)