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 ∈ L 1 (T 2 ) 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 ∈ L p (T 2 ), 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, Picar, Bessel, de La Vallée-Poussin, Rogosinski and Riesz summations.

Almost everywhere strong summability of cubic partial sums of d-dimensional Walsh-Fourier series

“…The results on strong summation and approximation of trigonometric Fourier series have been extended for several other orthogonal systems. For instance, concerning the Walsh system see Schipp [31,32,33], Fridli and Schipp [2,3], Leindler [20,21,22,23], Totik [36,37,38], Rodin [28], Weisz [41,42], Gabisonia [4], Goginava, Gogoladze [11].…”