Lebesgue points of double Fourier series and strong summability

Abstract

A general summability method of double Fourier series is given with the help of a double sequence θ(k,n). Under some conditions on θ we show that the Marcinkiewicz-θ-means of a function f∈L1(T2) converge to f at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points, for the so-called modified Lebesgue points of f∈Lp(T2), whenever 1<p<∞. The sufficient conditions of θ are proved for the Fejér, Abel and Cesàro summations. As an application we give simple proofs for the classical one-dimensional strong summability results of Hardy and Littlewood, Marcinkiewicz, Zygmund and Gabisoniya and generalize them for strong θ-summability. Some other special cases of the θ-summation are considered as well, such as the Weierstrass, Picard, Bessel, de La Vallée-Poussin, Rogosinski and Riesz summations.