On certain orthogonal polynomials, Nikolski- and Turán-type inequalities, and interpolatory properties of best approximants
type: Item
instance of: scholarly article
title: On certain orthogonal polynomials, Nikolski- and Turán-type inequalities, and interpolatory properties of best approximants
zbMATH Open document ID: 0786.41024
author: András Kroó
published in: Journal of Approximation Theory
publication date: 1994-04-21T00:00:00Z
review text: Let \(B_{n,p}(f)\) be the best \(L_ p\)-approximant of \(f\in C[-1,1]\) by polynomials of degree at most \(n\) (\(1<p\leq\infty\)). The main result of the paper: Assume that for a given subsequence \((n_ j)\), \(n_ 1<n_ 2<\dots\), such that \(f=B_{n_ j,p}(f)\) is zero-free on some interval \((a,b)\subseteq[-1,1]\). Then \(\limsup_{j\to\infty} n_{j+1}/n_ j>1\). This result constitutes a positive answer to a conjecture of \textit{E. B. Saff} and \textit{B. Shekhtman} [Indag. Math., New Ser. 1, No. 4, 489-498 (1990; Zbl 0723.41022)] concerning the lacunarity of bad subsequences of best approximation polynomials. For this purpose the author obtains general oscillatory properties of orthogonal polynomials (of independent interest) and fine asymptotic estimates.
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 205434
zbMATH Keywords: best \(L_ p\)-approximant
zbMATH Keywords: lacunarity of bad subsequences
zbMATH Keywords: best approximation polynomials
reviewed by: Ioan Şerb
MaRDI profile type: Publication
full work available at URL: jath.1993.1036
OpenAlex ID: W2031662585
DOI: 10.1006/JATH.1993.1036
description: scientific article; zbMATH DE number 205434