On the Bernstein conjecture in approximation theory
type: Item
instance of: scholarly article
title: On the Bernstein conjecture in approximation theory
zbMATH Open document ID: 0648.41013
DOI: 10.1007/BF01890040
author: Richard S. Varga
author: Amos J. Carpenter
published in: Constructive Approximation
publication date: 1985-01-01T00:00:00Z
review text: With \(E_{2n}(| x|)\) denoting the error of best uniform approximation to \(| x|\) by polynomials of degree at most 2n on the interval \([-1,+1]\), the famous Russian mathematician S. Bernstein in 1914 established the existence of a positive constant \(\beta\) for which \(\lim_{n\to \infty}2nE_{2n}(| x|)=:\beta.\) Moreover, by means of numerical calculations, Bernstein determined, in the same paper, the following upper and lower bounds for \(\beta\) : \(0.278<\beta <0.286\). Now, the average of these bounds is 0.282, which, as Bernstein noted as a ``curious coincidence,'' is very close to 1/(2\(\sqrt{\pi})=0.2820947917..\). This observation has over the years become known as the Bernstein Conjecture: Is \(\beta =1/(2\sqrt{\pi})?\) We show here that the Bernstein conjecture is false. In addition, we determine rigorous upper and lower bounds for \(\beta\), and by means of the Richardson extrapolation procedure, estimate \(\beta\) to approximately 50 decimal places.
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 4058128
zbMATH Keywords: error of best uniform approximation
zbMATH Keywords: Bernstein conjecture
zbMATH Keywords: Richardson extrapolation procedure
Wikidata QID: Q123119332
describes a project that uses: Algorithm 524
MaRDI profile type: Publication
cites work: Q5572451
cites work: Q4179737
cites work: Chebyshev Approximations for the Psi Function
cites work: Q4072022
cites work: Q5531436
cites work: Q5583224
cites work: Q3290778
full work available at URL: https://doi.org/10.1007/bf01890040
OpenAlex ID: W1967779414
description: scientific article; zbMATH DE number 4058128