Bernstein's comparison theorem and a problem of Braess
type: Item
instance of: scholarly article
title: Bernstein's comparison theorem and a problem of Braess
zbMATH Open document ID: 0533.41021
DOI: 10.1007/BF02188015
published in: Aequationes Mathematicae
publication date: 1981-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/136894
review text: As a partial generalization of the Bernstein comparison theorem, \textit{D. Braess} [Aequationes Math. 12, 80-81 (1975; Zbl 0328.41012)] proved the theorem: Let \(f,g\in C^{n+1}[a,b]\) and assume that (1) \(0\leq g^{(n+1)}(x)\leq f^{n+1}(x),\) \(x\in [a,b]\). Then (2) \(\min_{s\in {\mathcal S}_{n,k}}\| g-s\|_{\infty}\leq \min_{s\in {\mathcal S}_{n,k}}\| f-s\|_{\infty}.\) The author extended this theorem to the case where (3) \(| g^{(n+1)}(x)| \leq f^{(n+1)}(x),\) \(x\in [a,b]\) replaces assumption (1). In this case (2) is no longer valid. He demonstrates this fact by determining the precise upper bound on the best approximation of the functions g satisfying (3), by splines of degree n with k variable knots. It is also shown that interpolation at \(n+k+1\) fixed, distinct points by splines of degree n with k fixed knots (these depend upon f) gives the same upper bound as that which is obtained for the best approximation from \({\mathcal S}_{n,k}\) to the class of functions g satisfying (3). The author also characterizes the best constant in the degree of approximation from \({\mathcal S}_{n,k}\), of functions in the Sobolev space \(W_{\infty}\!^{n+1}\) which satisfy \(| g^{(n+1)}|_{\infty}\leq 1.\)
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A25
zbMATH DE Number: 3846148
zbMATH Keywords: Bernstein comparison theorem
zbMATH Keywords: best constant
zbMATH Keywords: degree of approximation
author: Allan Pinkus
reviewed by: László Leindler
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3846148