Unrestricted Cesàro summability of $d$-dimensional Fourier series and Lebesgue points

Abstract

We generalize the classical Lebesgue's theorem to multi-dimensional functions. We prove that the Cesàro means of the Fourier series of the multi-dimensional function $f\in L_1(\log L)^{d-1}(\mathbb{T}^d)\supset L_p(\mathbb{T}^d) (1<p<\infty)$ converge to $f$ at each strong Lebesgue point.