Cesàro summability and Lebesgue points of higher dimensional Fourier series

Abstract

<p style='text-indent:20px;'>We give four generalizations of the classical Lebesgue's theorem to multi-dimensional functions and Fourier series. We introduce four new concepts of Lebesgue points, the corresponding Hardy-Littlewood type maximal functions and show that almost every point is a Lebesgue point. For four different types of summability and convergences investigated in the literature, we prove that the Cesàro means <inline-formula><tex-math id="M1">\begin{document}$ \sigma_n^{\alpha}f $\end{document}</tex-math></inline-formula> of the Fourier series of a multi-dimensional function converge to <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula> at each Lebesgue point as <inline-formula><tex-math id="M3">\begin{document}$ n\to \infty $\end{document}</tex-math></inline-formula>.</p>