Polynomial approximation of functions on a quasi-smooth arc with Hermitian interpolation
type: Item
instance of: scholarly article
title: Polynomial approximation of functions on a quasi-smooth arc with Hermitian interpolation
zbMATH Open document ID: 1175.30005
author: H.-P. Blatt
author: V. V. Andrievskii
published in: Constructive Approximation
publication date: 2009-08-31T00:00:00Z
review text: Let \(L\) be a Jordan arc in the complex plane \(\mathbb C\) so that \(\Omega = \overline{\mathbb C} \setminus L\) is a simply connected domain. \(L\) is assumed to be quasi-smooth (in the sense of Lavrentiev), i. e., length \(L(z_1,z_2) \leq C_1|z_1 - z_2|\), \(C_1 = C(L) \geq 1\), for any \(z_1,\, z_2 \in L\), where \(L(z_1,z_2)\) is the subarc of \(L\) connecting the points \(z_1\) and \(z_2\). The goal of the paper is to give a theorem on polynomial approximation of functions \(f\) that are smooth on a quasi-smooth arc which simultaneously interpolate \(f\) and its derivative at certain points of \(L\). Some notation is need before the statement: \(D(z;\delta) = \{ \zeta : |z - \zeta| \leq \delta \}\), \(E_m(f,B) : = \inf_{f \in \mathbf{P}_m} \|f - p \|_B\), where \(B \subset L\), \(m \in \mathbb{N}_0\) and \(\mathbf{P}_m\) represent the class of polynomials of degree at most \(m\). The \(k\)th local modulus of smoothness is \[ \omega_{f,k,z,L}(\delta) : = E_{k -1}(f, L \cap D(z, \delta)) \] and the \(kth\) global modulus of smoothness of \(f\) on \(L\) is defined as \[ \omega_{f,k,L}(\delta) : = \sup_{z \in L}\omega_{f,k,z,L}(\delta). \] Let be \(\Omega = \overline{\mathbb{C}} \setminus L\), \(\Delta = \{ z : |z|> 1\}\) and \(\Phi\) the conformal map of \(\Omega\) onto \(\Delta \) such that \(\Phi(\infty) = \infty\) and \(\Phi '(\infty) = \lim_{z \to \infty} \frac{\Phi(z)}{z} > 0\). Set \(\Psi = \Phi^{-1}\) and note that \(\Phi\) can be continuously extended to the endpoints \(\zeta_1\) and \(\zeta_2\) of \(L\), and let be \(w_j : \Phi(\zeta_j)\), \(j= 1,2\). Let \(L_{\delta} = \{ \zeta : |\Phi(\zeta)| = 1 + \delta \}\). Put \(\Delta_1 : = \{ w: |w| > 1, \arg w_1 < \arg w < \arg w_2 \}\), \(\Delta_ 2 = \Delta \setminus \overline{\Delta_1}\), \(\Omega_j : = \Psi(\Delta_j)\), \(\rho_{\delta }^{(j)} = d(z, L_{\delta} \cap \Omega_j)\), and \(\rho_{\delta}^*(z) = \max(\rho_{\delta}^{(1)}(z), \rho_{\delta}^{(2)}(z))\). For \(0 < \delta < \rho\), let \(\Omega_{f,k, L}…
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 5600306
reviewed by: Fernando Pérez-González
MaRDI profile type: Publication
full work available at URL: s00365-008-9040-0
OpenAlex ID: W1977415307
cites work: APPROXIMATION CHARACTERIZATION OF CLASSES OF FUNCTIONS ON CONTINUA OF THE COMPLEX PLANE
cites work: Conformal invariants in constructive theory of functions of complex variable. Transl. from the Russian by D. N. Kravchuk
cites work: Simultaneous approximation and interpolation of functions on continua in the complex plane.
cites work: Q4358795
cites work: Pointwise approximation by algebraic polynomials
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: Interpolatory pointwise estimates for polynomial approximation
cites work: A theorem of gopengauz type with added interpolatory conditions
cites work: Q5685358
cites work: Strong and ordinary summability
cites work: Q4000499
cites work: Constructive characterization of classes of continuous functions of a set \(\mathfrak M\subset\mathbb{C}\) for the \(k-th\) modulus of continuity
cites work: Q5549176
cites work: Q4120973
cites work: A general direct theorem on approximation of functions of class \(C^r\) by algebraic polynomials with Hermitian interpolation
cites work: Constructive characteristic of continuous functions defined on smooth arcs
DOI: 10.1007/S00365-008-9040-0
description: scientific article; zbMATH DE number 5600306