Lebesgue points of two-dimensional Fourier transforms and strong summability
The recent literature of the Fourier analysis contains various studies concerning a general summation method called \(\theta\)-method. This method is generated by a single function \(\theta\) and includes Fejér, Riesz, Weierstrass, Abel and other known summation methods. The author proved in 2002 and 2011 that the Marcinkiewics means generated by the \(\theta\)-summation in the case \(f \in L_1(\mathbb{R})\) converge almost everywere to \(f\). In this paper, this result is generalized to Wiener amalgam spaces \(W(L_p,l_\infty)(\mathbb{R}^2)\), \(1\leq p\leq \infty\). The set of convergence is characterized by using the concepts of modified strong Lebesgue points in the case \(p=1\) and modified Lebesgue points in the case \(1<p<\infty\). Moreover, some classical results for the one-dimensional strong summability and for strong \(\theta\)-summability are generalized. Finally, the paper gives also some applications to various summability methods.
scientific article; zbMATH DE number 6515786
scientific article