$\ell_1$-summability and Lebesgue points of $d$-dimensional Fourier transforms

Abstract

‎The classical Lebesgue's theorem is generalized‎, ‎and it is proved that under some conditions on the summability function $\theta$‎, ‎the $\ell_1$-$\theta$-means of a function $f$ from the Wiener amalgam space $W(L_1,\ell_\infty)(\R^d)\supset L_1(\R^d)$ converge to $f$ at each modified strong Lebesgue point and thus almost everywhere‎. ‎The $\theta$-summability contains the Weierstrass‎, ‎Abel‎, ‎Picard‎, ‎Bessel‎, Fejér‎, ‎de La Vallée-Poussin‎, ‎Rogosinski‎, ‎and Riesz summations‎.