A new concept of Walsh-Lebesgue points is introduced for higher dimensions and it is proved that almost every point is a modified Walsh-Lebesgue point of an integrable function. It is shown that the Walsh-Fejér means σnf of a function f ∈ L1[0, 1)d converge to f at each modified Walsh-Lebesgue point, whenever n→∞ and n is in a cone. The same is proved for other summability means, such as for the Weierstrass, Abel, Picard, Bessel, Cesàro, de La Vallée-Poussin, Rogosinski and Riesz summations.
It is proved that the maximal operator of the triangular Cesàro means of a two-dimensional Fourier series is bounded from the periodic Hardy space to for all 2/(2+α)<p≦∞ and, consequently, is of weak type (1,1). As a consequence we obtain that the triangular Cesàro means of a function converge a.e. to f.