Complexity of weighted approximation over \(\mathbb{R}\)
type: Item
instance of: scholarly article
title: Complexity of weighted approximation over \(\mathbb{R}\)
zbMATH Open document ID: 0977.41009
published in: Journal of Approximation Theory
publication date: 2002-01-08T00:00:00Z
review text: This paper investigates approximation of univariate functions defined over the line. Assume that the \(r\)-th derivative of a function is bounded in a weighted \(L_w^p\) norm with a weight \(w\). Approximation algorithms use the values of a function and its derivatives up to the order \(r-1\). The worst case error of an algorithm is defined in a weighted \(L_u^q \) norm with a weight \(u\). This paper investigates the worst case (information) complexity of the weighted approximation problem, which is equal to the minimal number of functions and derivative evaluations needed to obtain the error \(\varepsilon\). This paper provides necessary and sufficient conditions in term of the weights \(w\) and \(u\), as well as the parameters \(r,p\), and \(q\) for the weighted approximation problem to have finite complexity. This paper also provides conditions which guarantee that complexity of weighted approximation is of the same order as the complexity of the classical approximation problem over a finite interval. Such necessary and sufficient conditions are also provided for a weighted integration problem since its complexity is equivalent to the complexity of the weighted approximation problem for \(q=1\).
reviewed by: Yingguang Shi
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 65Y20
Mathematics Subject Classification ID: 68Q25
zbMATH DE Number: 1443203
zbMATH Keywords: weighted approximation
zbMATH Keywords: complexity
zbMATH Keywords: approximation algorithm
author: Grzegorz W. Wasilkowski
author: H. Woźniakowski
MaRDI profile type: Publication
full work available at URL: jath.1999.3435
OpenAlex ID: W2010722609
cites work: Optimal integration of Lipschitz functions with a Gaussian weight
cites work: Q5533688
cites work: Asymptotically optimal weighted numerical integration
cites work: Deterministic and stochastic error bounds in numerical analysis
cites work: Q3344608
cites work: Q3673103
cites work: Q3993279
cites work: Explicit cost bounds of algorithms for multivariate tensor product problems
cites work: Weighted tensor product algorithms for linear multivariate problems
DOI: 10.1006/JATH.1999.3435
Recommended article: Complexity of weighted approximation over \(\mathbb{R}^d\)
Recommended article: Worst case complexity of weighted approximation and integration over \(\mathbb{R}^d\)
Recommended article: Average case complexity of weighted approximation and integration over \(\mathbb R_{+}\)
Recommended article: Weighted approximation in \(\mathbb{C} \)
Recommended article: Q4250299
Recommended article: Q4549522
Recommended article: Approximation by weighted polynomials in Rk
Recommended article: APPROXIMATION IN WEIGHTEDLPSPACES
Recommended article: Q5289135
Recommended article: Q3124068
description: scientific article; zbMATH DE number 1443203