Jackson-Bernstein theorem in strictly pseudoconvex domains in \({\mathbb{C}}^ n\)
type: Item
instance of: scholarly article
title: Jackson-Bernstein theorem in strictly pseudoconvex domains in \({\mathbb{C}}^ n\)
zbMATH Open document ID: 0683.32012
DOI: 10.1007/BF01889621
published in: Constructive Approximation
publication date: 1989-01-01T00:00:00Z
review text: Let \(\Omega\) be a bounded strictly pseudoconvex domain of \({\mathbb{C}}^ n\) with \(C^ 2\)-boundary. Denote by A(\(\Omega)\) the space of holomorphic function in the interior of \(\Omega\) and continuous on \({\bar \Omega}\) and by \(\Lambda^{\alpha}(\Omega)\), \(\alpha\in (0,1)\) the Hölder class of holomorphic functions on \({\bar \Omega}\). The main result: If the set of polynomials is dense in A(\(\Omega)\), then for any \(f\in \Lambda^{\alpha}(\Omega)\) there is a sequence of polynomials \(p_ N\), deg \(p_ N\leq N\) such that: \(| f(z)-p_ N(z)| <c_ f N^{-\alpha},\) \(z\in \partial \Omega.\) In the case \(\partial^{\beta}f\in \Lambda^{\alpha}(\Omega)\), for every multi-index \(\beta\) with \(| \beta |\) fixed, the author obtains analogous results.
Mathematics Subject Classification ID: 32E30
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 32A10
zbMATH DE Number: 4118707
zbMATH Keywords: complex approximation
zbMATH Keywords: Jackson-Bernstein theorem
zbMATH Keywords: strictly pseudoconvex domain
zbMATH Keywords: holomorphic function
author: N. A. Shirokov
reviewed by: Ioan Şerb
MaRDI profile type: Publication
cites work: Q4130476
cites work: Q3746093
cites work: Q4120973
description: scientific article; zbMATH DE number 4118707