The minimality properties of Chebyshev polynomials and their lacunary series
type: Item
instance of: scholarly article
title: The minimality properties of Chebyshev polynomials and their lacunary series
zbMATH Open document ID: 1078.33007
published in: Numerical Algorithms
publication date: 2005-04-25T00:00:00Z
review text: Let \(T_n, U_n, V_n\) and \(W_n\) be four kinds of Chebyshey polynomials \[ \begin{alignedat}{2} T_n(x)&= \text{cos} (n\theta),&\qquad U_n(x)&={{\sin ((n+1)\theta)}\over {\sin\theta}},\\ V_n(x)&={{\cos ((n+{1\over 2})\theta)}\over{\cos\left({\theta\over 2}\right)}},&\quad W_n(x)&={{\sin \left(\left(n+{1\over 2}\right)\theta\right)}\over{\sin\left( {\theta\over 2}\right)}} \end{alignedat} \] where \(x=\cos\theta\), \(0\leq \theta\leq\pi\). Define weight functions \(w_p(x)\), \(-1\leq x\leq 1\), for these polynomials as follows: if \(1\leq p<\infty\), then \[ w_p(x)=\begin{cases} (1-x^2)^{-{1\over 2}} &\text{ for }T_n,\\ (1-x^2)^{{p-1}\over 2} &\text{ for } U_n,\\ (1+x)^{{p-1}\over 2} (1-x)^{-{1\over 2}} &\text{ for } V_n,\\ (1-x)^{{p-1}\over 2} (1+x)^{-{1\over 2}}&\text{ for } W_n \end{cases} \] and if \(p=\infty\) \[ w_\infty(x)=\lim_{p\to +\infty} (w_p(x))^{{1\over p}}. \] The author proves the following theorems that extend a number of known results on Chebyshev polynomials. Chebyshev's equioscillation theorem: Let \(p_n\) be a polynomial, \(\deg p=n\), and \(f\) be continuous. For four cases of \(w_{\infty}(x)\), the norm \[ \| f-p_n\|=\max_{-1\leq x\leq 1} |w_\infty(x) (f(x)-p_n(x))| \] is minimised if and only if \(w_\infty(f-p_n)\) attains its maximum magnitude with alternating signs on at least \(n+2\) consecutive points of \([-1, 1]\). The \(L_p\) minimality property: The monic polynomials which are corresponding \(T_n, U_n, V_n, W_n\) are the best \(L_p\) approximations to zero on \([-1, 1]\) with respect to their \(w_p(x)\), \(1\leq p\leq \infty\). A sufficient interpolation condition for weight \(L_1\) approximation: Let \(f\) be a continuous function on \([-1, 1]\). A polynomial \(p_n\), \(\deg p_n=n\), is the best \(L_1\) approximation of \(f\) with respect to \(w_1(x)\) if zeros of \(f-p_n\) coincide with zeros of the relevant \(n\)-th Chebyshev polynomial corresponding to \(w_1(x)\). The best \(L_p\) approximation by partial sums of lacunary series: L…
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 42C15
zbMATH DE Number: 2161187
zbMATH Keywords: Chebyshev polynomials
zbMATH Keywords: lacunary Chebyshev series
zbMATH Keywords: Chebyshev interpolation
zbMATH Keywords: best and near-best polynomial approximation
author: J. C. Mason
reviewed by: Aleksey A. Dovgoshey
MaRDI profile type: Publication
cites work: Q5543516
cites work: Best and near-best \(L_ 1\) approximations by Fourier series and Chebyshev series
cites work: Q3900444
description: scientific article; zbMATH DE number 2161187