The Gibbs phenomenon for \(L_{\text{loc}}^1\) kernels
The authors prove two theorems on the Gibbs phenomenon for multiple Fourier integrals in \(\mathbb{R}^n\), by reducing the problem to the one-dimensional case. In a third theorem they specify their results in the case of the Bochner-Riesz means of order \(\alpha> (n-3)/2\) of functions \(f: \mathbb{R}^n\to \mathbb{R}\) vanishing in \(\mathbb{R}^n\setminus C\) and smooth enough in \(C\), where the boundary of \(C\) is a smooth simple surface. The case when \(\alpha\leq(n- 1)/2\), the critical index, is especially remarkable.
scientific article; zbMATH DE number 1356335