Best approximation and saturation on domains bounded by curves of bounded rotation
type: Item
instance of: scholarly article
title: Best approximation and saturation on domains bounded by curves of bounded rotation
zbMATH Open document ID: 0952.41015
author: I. Bruj
author: Gerald Schmieder
published in: Journal of Approximation Theory
publication date: 1999-12-20T00:00:00Z
review text: Let \(G\) be a Jordan domain with a boundary curve of bounded rotation. By \(A(\overline G)\) is denoted the class of functions that are holomorphic in \(G\) and continuous on the closure \(\overline G\). Let further \(p_m(\overline G,z)\) be the Faber polynomial of degree \(m\) for the set \(\overline G\). For functions \(f\in A(\overline G)\) having the so-called Faber derivative \(\varphi =F^rf,r\in N\cup\{0\}\), satisfying the Hőlder condition of order \(\alpha ,0<\alpha\leq 1,\) and for a matrix \(\mu =(\mu _m^n)\), the folloving means are considered \[ U_n^{\mu} (f,\overline G,z)=a_0(f,\overline G)+\sum _{m=1}^{n}(im)^{-r}a_{m} (\varphi ,\overline G)\mu _m^np_m(\overline G,z),\quad z\in \mathbb{C}. \] Here \(a_m\) are the Faber coefficients of the function in the parentheses. Under certain restrictions on the matrix \(\mu \) a relation is proved which contains full information about the order of saturation. The applications of this result relative to the partial sums, Fej\(\acute e\)r, Riesz and Rogosinski means are given and the saturation order of these methods are established.
Mathematics Subject Classification ID: 41A40
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 1356336
zbMATH Keywords: holomorphic functions
zbMATH Keywords: saturation
zbMATH Keywords: Faber polynomials
zbMATH Keywords: best approximation
zbMATH Keywords: means of Faber series
reviewed by: D. K. Ugulava
MaRDI profile type: Publication
full work available at URL: jath.1998.3335
OpenAlex ID: W2007660462
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DOI: 10.1006/JATH.1998.3335
description: scientific article; zbMATH DE number 1356336