A 3-dimensional Bergman kernel method with applications to rectangular domains
The Bergman kernel method (BKM) is a method for approximating the mapping \(f\) which maps conformally a bounded simply connected domain \(\Omega\subset\mathbb{C}\) onto the unit disc \(D\subset\mathbb{C}\) such that \(f(0)=0\) and \(f'(0)>0\). This method strongly bases on approximations of the kernel function of \(\Omega\) and its reproducing property and on complete systems of functions for \(L^2(\Omega)\). The goal of the authors is to generalize this method to \(\mathbb{R}^3\) and to approximate mappings \(f:\mathbb{R}^3\to\mathbb{R}^3\). Here they go back to the Fueter polynomials which depend on \(z=x_0+e_1x_1+e_2x_2+e_3x_3\). Generalizing BKM the authors propose a strategy which approximates a given \(f\) by omitting the resulting \(e_3\)-part after following the standard procedure of BKM. An analytic verification is missing. Some numerical experiments hint to applicability of the modified BKM in \(\mathbb{R}^3\).
scientific article; zbMATH DE number 5015119