Extension of vector-valued holomorphic and meromorphic functions
type: Item
instance of: scholarly article
title: Extension of vector-valued holomorphic and meromorphic functions
zbMATH Open document ID: 1091.46022
author: Enrique Jordá
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2005-07-05T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bbms/1113318125
review text: Let \(\emptyset \not= \Omega_1 \subseteq \Omega_2 \subseteq \mathbb{C}\) be any domains and let \(E\) be a locally convex space. Let \(f:\Omega_1 \to E\) be any holomorphic or meromorphic function such that all compositions \(u \circ f:\Omega_1 \to E \to \mathbb{C}\) by linear functionals \(u \in E'\) admit a holomorphic or meromorphic extension to \(\Omega_2\), respectively. Does there exist a holomorphic or meromorphic extension \(\tilde{f}:\Omega_2 \to E'\) of \(f\)? It is well known that the answer is yes for all Banach spaces \(E\) [see e.g., \textit{L. M. Hai, N. V. Khue} and \textit{N. T. Nga}, Colloq. Math. 64, No. 1, 65--70 (1993; Zbl 0824.46052)]. In the paper under review the author studies the problem for locally convex spaces. In the holomorphic case the answer is yes for all locally complete spaces \(E\), and in the meromorphic case the answer is yes under the additional assumption that \(E\) does not contain \(\mathbb{C}^{\mathbb{N}}\) and that \(E'_{\beta}\) is suprabarrelled or that \(E\) is a barrelled complete Schwartz space. As a main tool it is used that for locally complete spaces \(E\) not containing \(\mathbb{C}^{\mathbb{N}}\) the formula \(M(\Omega, E)=M(\Omega) \varepsilon E\) holds true.
reviewed by: Heinz Junek
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46A03
Mathematics Subject Classification ID: 46E05
Mathematics Subject Classification ID: 30D30
Mathematics Subject Classification ID: 30G30
zbMATH DE Number: 2186499
zbMATH Keywords: holomorphic extension
zbMATH Keywords: meromorphic extension
zbMATH Keywords: \(\varepsilon\)-product
MaRDI profile type: Publication
Recommended article: Q4508521
Recommended article: Extension of Holomorphic Mappings from E to E"
Recommended article: Q4307188
Recommended article: Q4252485
Recommended article: A Hahn-Banach theorem for holomorphic mappings on localy convex spaces
Recommended article: Meromorphic functions with values in a Fréchet space and linear topological invariant (DN)
Recommended article: Locally Bounded Sets of Holomorphic Mappings
Recommended article: Q4697536
Recommended article: Extension of Entire Functions on Nuclear Locally Convex Spaces
Recommended article: Weak holomorphic extension of Fréchet-valued functions form compact subsets of \(\mathbb{C}^n\)
description: scientific article; zbMATH DE number 2186499