Local Lipschitz constants
type: Item
instance of: scholarly article
title: Local Lipschitz constants
zbMATH Open document ID: 0552.41018
DOI: 10.1016/0021-9045(85)90148-0
author: Myron S. Henry
author: Terry D. Lenker
author: James R. Angelos
author: Edwin H. jun. Kaufman
published in: Journal of Approximation Theory
publication date: 1985-01-01T00:00:00Z
review text: Let X be a closed subset of \(I=[-1,1]\). For \(f\in C[X]\), the local Lipschitz constant is defined to be \(\lambda_{n\delta}(f)=\sup \{\| B_ n(f)-B_ n(g)\| /\| f-g\|:0<\| f-g\| \leq \delta \}\), where \(B_ n(g)\) is the best approximation in the sup norm to g on X from the set of polynomials of degree at most n. It is shown that under certain assumptions the norm of the derivative of the best approximation operator at f is equal to the limit as \(\delta\) \(\to 0\) of the local Lipschitz constant of f, and an explicit expression is given for this common value. Then possibly very different, characterizations of local and global Lipschitz constants are also considered.
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 3879467
zbMATH Keywords: local Lipschitz constant
zbMATH Keywords: best approximation operator
MaRDI profile type: Publication
full work available at URL: 0021-9045(85)90148-0
OpenAlex ID: W1980579897
cites work: On Lipschitz conditions, strong unicity and a theorem of A. K. Cline
cites work: Q3662047
cites work: Q5543516
cites work: Lipschitz conditions on uniform approximation operators
cites work: A uniform constant of strong uniqueness on an interval
cites work: Q3344652
cites work: Q3697549
cites work: Q4108735
cites work: On extremal sets and strong unicity constants for certain \(C^{\infty}\) functions
cites work: Lipschitz and strong unicity constants for changing dimension
cites work: Uniform Lipschitz constants on small intervals
cites work: The Lipschitz constant of the operator of best approximation
cites work: Differential properties of the operator of best approximation
cites work: The continuity of best approximations
cites work: Q5583224
cites work: Q3937129
description: scientific article; zbMATH DE number 3879467