Discretization for uniform polynomial approximation
type: Item
instance of: scholarly article
title: Discretization for uniform polynomial approximation
zbMATH Open document ID: 0532.41008
DOI: 10.1016/0021-9045(84)90117-5
published in: Journal of Approximation Theory
publication date: 1984-01-01T00:00:00Z
review text: Let P be the polynomial of degree less than or equal to n which is the best approximation to a given f in C[-1,1]. An approximation to P can be computed by choosing a finite subset F of [-1,1] and calculating the polynomial \(P_ F\), of degree less than or equal to n, which best approximates f on F. Then if \(| F|\) is small, estimates show that the discretization error, as measured by \(\| P-P_ F\|\), is also small. A classical choice for the set F of m points is (1) \(\{\) cos((2j- 1)\(\pi\) /2m): \(j=1,2,...,m\}\). A natural formulation of this discretization problem is developed which leads to a specific criterion for the choice of points in F. It is shown that, by this criterion, the choice of points in (1) is asymptotically best, but not best.
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 3844302
zbMATH Keywords: discretization
zbMATH Keywords: Markov's inequality
zbMATH Keywords: Bernstein's inequality
author: R. Whitley
MaRDI profile type: Publication
full work available at URL: 0021-9045(84)90117-5
OpenAlex ID: W1981401979
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description: scientific article; zbMATH DE number 3844302