CESARO SUMMABILITY AND LEBESGUE POINTS OF HIGHER DIMENSIONAL FOURIER SERIES

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 Cesaro means sigma(alpha)(n) f of the Fourier series of a multi-dimensional function converge to f at each Lebesgue point as n -> infinity.