Q1078375
type: Item
instance of: scholarly article
title: Meilleure approximation polynomiale et croissance des fonctions entières sur certaines variétés algébriques affines. (Best polynomial approximation and growth of entire functions on certain affine algebraic varieties)
zbMATH Open document ID: 0596.32025
author: Ahmed Zeriahi
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/74758
review text: Let K be a polynomially convex compact set in \({\mathbb{C}}^ n\) and \(V_ K\) its logarithmic extremal potential in \({\mathbb{C}}^ n\). Suppose that K is regular (e.g. \(V_ K\) is continuous). Then we construct for every holomorphic function f on K a sequence \((P_{\ell})_{\ell \geq 1}\) of polynomials in n complex variables with deg (P\({}_{\ell})\leq \ell\) such that: \(\forall \vartheta >1\), \(\exists c_{\vartheta}>0\), \(\exists N_{\vartheta}>1\) such that \[ \max_{z\in K} | f(z)- P_{\ell}(z)| \leq [c_{\vartheta}(1+r)^{N_{\vartheta}}/(r-\quad 1)^{2n-1}]\cdot \| f\|_{{\bar \Omega}_{r\vartheta}}/r^{\ell} \] for every \(r>1\), \(\ell \geq 1\), f holomorphic on \({\bar \Omega}{}_{r\vartheta}\) where \(\Omega_{r\vartheta}=\{z\in {\mathbb{C}}^ n: \exp V_ K(z)<r\vartheta \}.\) The constants \(c_{\vartheta},N_{\vartheta}\) don't depend on r,\(\ell\) and f. Moreover, \(P_{\ell}\) depends linearly on f and is given by an integral operator obtained from the theory of integral formula with weight kernels of Khenkin-Ramirez type developed recently by Berndtsson and Anderson. The above estimates are then used to extend to several complex variables a classical result of S. N. Bernstein relating the analytic continuation of a continuous function on K by an entire function of prescribed order and type with its best polynomial approximation on K.
Mathematics Subject Classification ID: 32E30
Mathematics Subject Classification ID: 32A22
Mathematics Subject Classification ID: 32E20
Mathematics Subject Classification ID: 31C15
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 30A05
Mathematics Subject Classification ID: 31C10
Mathematics Subject Classification ID: 47Gxx
Mathematics Subject Classification ID: 32A30
zbMATH DE Number: 3959941
zbMATH Keywords: polynomially convex
zbMATH Keywords: logarithmic extremal potential
zbMATH Keywords: integral formulas with weight kernels of Henkin-Ramirez type
zbMATH Keywords: order
zbMATH Keywords: type
zbMATH Keywords: best polynomial approximation
zbMATH Keywords: growth of entire functions
zbMATH Keywords: analytic continuation of continuous functions by entire functions
MaRDI profile type: Publication
OpenAlex ID: W2332739656
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DOI: 10.5802/AIF.1087
published in: Annales de l’institut Fourier
label: Meilleure approximation polynomiale et croissance des fonctions entières sur certaines variétés algébriques affines. (Best polynomial approximation and growth of entire functions on certain affine algebraic varieties)
description: scientific article; zbMATH DE number 3959941