Interpolatory properties of best rational \(L_ 1\)-approximations
type: Item
instance of: scholarly article
title: Interpolatory properties of best rational \(L_ 1\)-approximations
zbMATH Open document ID: 0676.41018
DOI: 10.1007/BF02075450
author: András Kroó
author: Franz Peherstorfer
published in: Constructive Approximation
publication date: 1988-01-01T00:00:00Z
review text: The problem concerns the distance between interpolation points of the best approximation to a continuous function. For example, for approximation by polynomials of degree n on [-1,1] the maximal distance between interpolating points is of the order O(1/n). This same question is asked of the interpolating points from \(R_{m,n}\)- the quotients of polynomials of degree m in the numerator and n in the denominator. They find for example, that if r is a best \(L_ 1\) approximation to f on [-1,1] and (a,b) is contained in [-1,1] and arccos(a)-arccos(b) is less than \(\pi /(n-m+2)\) then r agrees with f on some point in (a,b). Furthermore this bound is, in general, best possible. Their other results involve functions defined on subsets of \(R^ 2\), and they have several pertinent examples. In general terms, they find that the greater the quantity m-n, the more densely distributed are the interpolating points.
Mathematics Subject Classification ID: 41A20
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 4107360
zbMATH Keywords: best \(L_ 1\) approximation
zbMATH Keywords: examples
MaRDI profile type: Publication
cites work: The Lipschitz constant of the operator of best approximation
cites work: Chebyshev Rank in L 1 -Approximation
cites work: On the representation of extremal functions in the \(L^1\)-norm
cites work: Nonlinear \(L_1\) approximation of smooth functions
cites work: Critical points in nonlinear \(L^ 1\) approximation
full work available at URL: https://doi.org/10.1007/bf02075450
OpenAlex ID: W1994199810
description: scientific article; zbMATH DE number 4107360