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 ∈ L1(log L)d−1(Td) ⊃ Lp(Td) (1 < p ≤ ∞) converge to f at each strong Lebesgue point.