Lebesgue Points of Two-Dimensional Fourier Transforms and Strong Summability

Abstract

We introduce the concept of modified strong Lebesgue points and show that almost every point is a modified strong Lebesgue point of \(f\) from the Wiener amalgam space \(W(L_1,\ell _\infty )({\mathbb {R}}^2)\). A general summability method of 2D Fourier transforms is given with the help of an integrable function \(\theta \). Under some conditions on \(\theta \) we show that the Marcinkiewicz-\(\theta \)-means of a function \(f\in W(L_1,\ell _\infty )({\mathbb {R}}^2)\) 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 \(f\in W(L_p,\ell _\infty )({\mathbb {R}}^2)\), whenever \(1<p<\infty \). As an application we generalize the classical 1D strong summability results of Hardy and Littlewood, Marcinkiewicz, Zygmund and Gabisoniya for \(f\in W(L_1,\ell _\infty )({\mathbb {R}})\) and for strong \(\theta \)-summability. Some special cases of the \(\theta \)-summation are considered, such as the Weierstrass, Abel, Picar, Bessel, Fejer, de La Vallee-Poussin, Rogosinski and Riesz summations.