Strong convergence of approximate identities and Bourgain points of bounded functions
The author first describes a class \(\Phi\) such that any bounded function is strongly (in the sense of Lim) approximated by the convolutions \(u^f_\Phi(x,y)\) at each of its \(B\)-points \(x\). Next, he describes the kernels \(\Phi\) such that \(\text{Lim}_{y\to -} u^f_\Phi(x,y)\) for a real-valued \(f\in C(\mathbb{R})\cap L^\infty(\mathbb{R})\) does not exist at all points \(x\in\mathbb{R}\). The paper also contains a geometrical description of \(B\)-pints of Cantor-type perfect sets. A multidimensional analogue of a result obtained in this paper is also mentioned.
scientific article; zbMATH DE number 5656100