Limits of strong unicity constants for certain \(C^{\infty}\) functions
type: Item
instance of: scholarly article
title: Limits of strong unicity constants for certain \(C^{\infty}\) functions
zbMATH Open document ID: 0564.41013
DOI: 10.1007/BF01958028
published in: Acta Mathematica Hungarica
publication date: 1984-01-01T00:00:00Z
review text: Let C[-1,1] denote the space of all continuous real-valued functions on the interval [-1,1] endowed with the uniform norm, and let \(\Pi_ n\) be the subspace consisting of all polynomials of degree at most n. It is known that for each \(f\in C[-1,1]\) with best approximation \(B_ n(f)\) from \(\Pi_ n\) there is a smallest positive constant \(M_ n(f)\), called strong unicity constant, such that \(\| p-B_ n(f)\| \leq M_ n(f)[\| f-p\| -\| f-B_ n(f)\|]\) for all \(p\in \Pi_ n\). The authors' aim is to find classes of functions f for which \(\lim_{n\to \infty}M_ n(f)/n\) can be determined. Two such classes are pointed out. Firstly, it is proved that \(\lim_{n\to \infty}M_ n(f)/n=2(a+1)^{1/2}/(a-1)^{1/2}\) when f is the rational function defined by \(f(x)=1/(a-x)\) with \(a\geq 2\). Secondly, it is shown that for each f belonging to a certain class of continuously differentiable non- rational functions the equality \(\lim_{n\to \infty}M_ n(f)/n=2\) holds.
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 3899487
zbMATH Keywords: strong unicity constant
author: Myron S. Henry
author: John Swetits
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01958028
OpenAlex ID: W1973139807
description: scientific article; zbMATH DE number 3899487