A 3-dimensional Bergman kernel method with applications to rectangular domains
type: Item
instance of: scholarly article
title: A 3-dimensional Bergman kernel method with applications to rectangular domains
zbMATH Open document ID: 1099.30003
author: S. Bock
author: K. Gürlebeck
author: Maria Irene Falcão
author: Helmuth R. Malonek
published in: Journal of Computational and Applied Mathematics
publication date: 2006-03-24T00:00:00Z
review text: 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 \(\mathbb{R}^3\).
reviewed by: Michael Reissig
Mathematics Subject Classification ID: 30C30
Mathematics Subject Classification ID: 65E05
Mathematics Subject Classification ID: 30B60
zbMATH DE Number: 5015119
zbMATH Keywords: Fueter polynomials
zbMATH Keywords: quaternions
zbMATH Keywords: numerical package
MaRDI profile type: Publication
full work available at URL: j.cam.2005.04.058
OpenAlex ID: W2032768807
Wikidata QID: Q58197970
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DOI: 10.1016/J.CAM.2005.04.058
description: scientific article; zbMATH DE number 5015119