Abstract. We introduce a new concept of Lebesgue points, the so-called $\omega $-Lebesgue points, where $\omega >0$. As a generalization of the classical Lebesgue's theorem, we prove that the Cesàro means $\sigma _n^{a}f$ of the Fourier series of a multi-dimensional function $f\in L_1(\T ^d)$ converge to $f$ at each $\omega $-Lebesgue point $(0<\omega <\alpha )$ as $n\to \infty $.

DOI: 10.14232/actasm-021-614-3

AMS Subject Classification (1991): 42B08, 42A38, 42A24, 42B25

Keyword(s): Cesàro summability, Hardy--Littlewood maximal function, Lebesgue points

received 14.1.2021, revised 29.8.2021, accepted 31.8.2021. (Registered under 114/2021.)