Tchebycheff polynomials on a disk.
type: Item
instance of: scholarly article
title: Tchebycheff polynomials on a disk.
zbMATH Open document ID: 1036.41006
author: D. Massart
published in: Journal of Computational and Applied Mathematics
publication date: 2004-01-26T00:00:00Z
review text: There is a well-known characterization of the Chebychev polynomials of the first kind \(T_n\) \[ \| x^n-q_{n-1}\| _{[-1,1]}\geq 2^{1-n}\| T_n\| _{[-1,1]}, \] where \({\| \cdot\| _{[-1,1]}}\) is the uniform norm on the segment \({[-1,1]}\), \(q_n\) is an arbitrary polynomial of degree\({\;\leq n-1}\). Let \(\Pi^2_n\) be the space of polynomials in two variables with total degree\({\;\leq n}\). Let \(D\) be the unit disk on the plane. The authors define the Chebychev polynomials \(T_{n,m}\) on \(D\) and show that \[ \| x^m y^n-q_{m,n}(x,y)\| _{D}\geq 2^{1-n-m}\| T_{m,n}\| _{D}, \] where \({q_{m,n}\in\Pi^2_{n+m-1}}\). The Chebychev polynomials \(T_{n,m}\) are expressed explicitly by the classical Chebychev polynomials of first and second kind.
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A52
zbMATH DE Number: 2032631
zbMATH Keywords: Chebychev polynomials
zbMATH Keywords: uniform norm
zbMATH Keywords: best approximation
MaRDI profile type: Publication
full work available at URL: j.cam.2003.05.004
OpenAlex ID: W2073334921
cites work: Q5534404
cites work: Q5531436
cites work: Tchebycheff polynomials on a triangular region
cites work: Some new characterizations of the Chebyshev polynomials
DOI: 10.1016/J.CAM.2003.05.004
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description: scientific article; zbMATH DE number 2032631