Kergin interpolants of holomorphic functions
type: Item
instance of: scholarly article
title: Kergin interpolants of holomorphic functions
zbMATH Open document ID: 0904.32011
published in: Constructive Approximation
publication date: 1999-01-21T00:00:00Z
review text: Let \(D\subset\mathbb{C}^n\) be a \(\mathbb{C}\)-convex domain, that is, its intersection with each complex line \(\ell\) is a simply connected domain in \(\ell\). Let \(\{a_{nm}\}\), \(n= 1,2,\dots\), \(m= 1,2,\dots, n\) be an array of points in a compact set \(K\subset D\). Then, it is shown that for a function \(f\) holomorphic in \(D\) there exists a polynomial \(P_n\) in \(\mathbb{C}^n\) of degree \(\leq n\) such that \(P(a_{nm}\}= f(a_{nm})\), \(m= 1,2,\dots, n\). An integral formula for the error \(f(z)- P_n(z)\) is proved. It is a generalization of the one-dimensional formula of Hermite. This formula is used to prove the main result of the paper: a theorem giving conditions which imply the uniform convergence of \(P_n(f)\) to \(f\) on \(K\). These conditions are given in terms of potential theory.
Mathematics Subject Classification ID: 32E30
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 32A10
zbMATH DE Number: 1129649
zbMATH Keywords: interpolation of a function
zbMATH Keywords: approximation
zbMATH Keywords: formula of Hermite
author: Thomas Bloom
author: Jean-Paul Calvi
MaRDI profile type: Publication
full work available at URL: s003659900059
OpenAlex ID: W2036082589
DOI: 10.1007/S003659900059
description: scientific article; zbMATH DE number 1129649