Triangular summability of two-dimensional Fourier transforms

We consider the triangular summability of two-dimensional Fourier transforms, and show that the maximal operator of the triangular-??-means of a tempered distribution is bounded from H _p (?²) to L _p (?²) for all 2/(2 + ??) < p ?? ??; consequently, it is of weak type (1,1), where 0 < ?? ?? 1 is depending only on ??. As a consequence, we obtain that the triangular-??-means of a function f ?? L ₁(?²) converge to f a.e. Norm convergence is also considered, and similar results are shown for the conjugate functions. Some special cases of the triangular-??-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, de la Vallée-Poussin, Rogosinski, and Riesz summations.