Marcinkiewicz summability of Fourier series, Lebesgue points and strong summability

Abstract

Under some conditions on \({\theta}\), we characterize the set of convergence of the Marcinkiewicz-\({\theta\mbox{-}}\)means of a function \({f \in L_1(\mathbb{T}^d)}\). More exactly, the \({\theta\mbox{-}}\)means 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 \({f \in L_p(\mathbb{T}^d)}\), whenever \({1 < p < \infty}\). The \({\theta\mbox{-}}\)summability includes the Fejer, Abel, Cesaro and some other 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 \({\theta\mbox{-}}\)summability.