Constructive theory of analytic functions on a quasidisk
type: Item
instance of: scholarly article
title: Constructive theory of analytic functions on a quasidisk
zbMATH Open document ID: 1010.30023
DOI: 10.1007/BF03320983
published in: Computational Methods and Function Theory
publication date: 2003-05-07T00:00:00Z
review text: Let \(K\subset\mathbb C\) be a compact set with connected complement. Denote by \(\mathcal A(K)\) the class of all functions continuous on \(K\) and analytic in the interior of \(K\). Let \(\mathcal P_n\) be the class of complex polynomials of degree at most \(n\in\mathbb N\). For \(f\in\mathcal A(K)\) and \(n\in\mathbb N\), denote by \(E_n(f,K) :=\inf\{\|f-p\|_K\), \(p\in\mathcal P_n\} \) the best uniform approximations of \(f\) on \(K\). The following two problems are fundamental for the constructive theory of functions of a complex variable: (1) describe the rate of polynomial approximation of all functions \(f\in\mathcal A(K)\) with given smoothness properties; (2) describe all functions \(f\in\mathcal A(K)\) with given rate of decrease of \(E_n(f,K)\) as \(n\to\infty\). In this paper, the author considers the case of functions on a closed quasidisk \(\overline G\), i.e., a domain bounded by a quasiconformal curve \(\partial G\). Direct and inverse theorems, establishing a connection between the rate of polynomial approximation of the function \(f\) on the boundary \(\partial G\) and its smoothness properties, are proved. The main idea of these theorems is that both problems can be treated in the same way.
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 1833062
zbMATH Keywords: polynomial approximation on quasidisk
zbMATH Keywords: constructive description
zbMATH Keywords: rate of polynomial approximation
zbMATH Keywords: smoothness properties
author: V. V. Andrievskii
reviewed by: Eleonora A. Storozhenko
MaRDI profile type: Publication
cites work: Q5512337
cites work: Approximation of functions in domains with a quasiconformal boundary
cites work: Conformal invariants in constructive theory of functions of complex variable. Transl. from the Russian by D. N. Kravchuk
cites work: APPROXIMATION OF SOLUTIONS OF THE EQUATION $ \overline\partial^jf=0$, $ j\geq1$, IN DOMAINS WITH QUASICONFORMAL BOUNDARY
cites work: Simultaneous approximation and interpolation of functions on continua in the complex plane.
cites work: CONFORMAL MAPPINGS AND THE APPROXIMATION OF ANALYTIC FUNCTIONS IN DOMAINS WITH A QUASICONFORMAL BOUNDARY
cites work: On the theory of approximation of functions on closed sets of the complex plane (with reference to a problem of S. M. Nikol'skii)
cites work: Q5569283
cites work: Q5685358
cites work: APPROXIMATION OF MONOTONE FUNCTIONS BY MONOTONE POLYNOMIALS
cites work: Q5607484
cites work: Q4120973
cites work: Polynomial approximations and smoothness moduli of functions in regions with a quasiconformal boundary
Recommended article: Function theory on a \(q\)-analog of complex hyperbolic space
Recommended article: On a class of analytic functions in the unit disk
Recommended article: Q3491861
Recommended article: Quasianalytic classes of functions on a plane
Recommended article: Q3688620
Recommended article: Q4734073
Recommended article: Q4029742
Recommended article: Q3362974
Recommended article: Constructive description of analytic functions of the Hölder class
Recommended article: Construction of analytic discs and regularity of functions holomorphic at the boundary
description: scientific article; zbMATH DE number 1833062