Interpolatory properties of best \(L_ 2\)-approximants
type: Item
instance of: scholarly article
title: Interpolatory properties of best \(L_ 2\)-approximants
zbMATH Open document ID: 0723.41022
DOI: 10.1016/0019-3577(90)90017-H
published in: Indagationes Mathematicae. New Series
publication date: 1990-01-01T00:00:00Z
review text: Let f be a continuous function and \(S_ n\) be the polynomial of degree at most n of best \(L_ 2(\mu)\)-approximation to f on [-1,1]. Let \(Z_ n(f)=\{x\in [-1,1]\); \(f(x)-S_ n(x)=0\}\), i.e. \(S_ n(x)\) interpolates f(x) at the points of \(Z_ n(f)\). The main result of this paper is that \(\cup_{n\in N}Z_ n(f)\) is dense in [-1,1] under mild conditions on the measure \(\mu\). This answers a question posed independently by A. Kroo and V. M. Tikhomiroff. Some related results for the space \(L_ p(\mu)\) (1\(\leq p\leq \infty)\) are also proved.
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 4192340
zbMATH Keywords: best \(L_ 2\)-approximants
author: E. B. Saff
author: Boris Shekhtman
reviewed by: D. Zarnadze
MaRDI profile type: Publication
cites work: The Density of Alternation Points in Rational Approximation
cites work: Q3202618
cites work: The distribution of extreme points in best complex polynomial approximation
cites work: Interpolatory properties of best \(L_ 1\)-approximants
cites work: Interpolatory properties of best rational \(L_ 1\)-approximations
cites work: The Density of Extreme Points in Complex Polynomial Approximation
cites work: Distribution of Alternation Points in Uniform Polynomial Approximation
cites work: Behavior of best \(L_ p\) polynomial approximants on the unit interval and on the unit circle
cites work: Strong and Weak Convergence of Orthogonal Polynomials
cites work: On the Zeros of Jacobi Polynomials P ( | alpha n , | beta n ) n (X)
cites work: Orthogonal polynomials
cites work: Q3034288
cites work: Weighted Polynomial Approximation of Analytic Functions
cites work: Q5595759
cites work: Q4743225
Recommended article: Interpolatory properties of best \(L_ 1\)-approximants
Recommended article: Interpolatory properties of best rational \(L_ 1\)-approximations
Recommended article: Interpolation and approximation in \(L_{2}(\gamma )\)
Recommended article: Q3814175
Recommended article: Q5748363
Recommended article: Q3550059
Recommended article: Q5275309
Recommended article: Q3202613
Recommended article: Q3743764
Recommended article: Best interpolatory approximation in normed linear spaces
description: scientific article; zbMATH DE number 4192340