Triangular summability and Lebesgue points of 2-dimensional Fourier transforms

Abstract

We consider the triangular $\theta$-summability of $2$-dimensional Fourier transforms. Under some conditions on $\theta$, we show that the triangular $\theta$-means of a function $f$ belonging to the Wiener amalgam space $W(L_1,{\ell }_{\infty })\left({\mathbb{R}}^2\right)$ converge to $f$ at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points for the so-called modified Lebesgue points of $f\in W(L_p,{\ell }_{\infty })\left({\mathbb{R}}^2\right)$ whenever $1<p<\infty$. Some special cases of the $\theta$-summation are considered, such as the Weierstrass, Abel, Picard, Bessel, Fejér, de La Vallée-Poussin, Rogosinski, and Riesz summations.