Convergence of summability means of higher dimensional Fourier series and Lebesgue points

Abstract

We introduce a new concept of Lebesgue points for higher dimensional functions. Every continuity point is a Lebesgue point and almost every point is a Lebesgue point of an integrable function. Given a strictly increasing continuous function\(\delta\), we prove that the Fejér or Cesàro means\(\sigma_n^{\alpha}f\) of the Fourier series of a two-dimensional function \(f\in L_1(\mathbb{T}^2)\) converge to \(f\) at each Lebesgue point as \(n\to \infty\) and n is in the cone around the graph of \(\delta\). We also prove this result for higher dimensional functions and for other summability means. This is a generalization of the classical one-dimensional Lebesgue’s theorem for the Fejér means.