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Behavior of alternation points in best rational approximation

Behavior of alternation points in best rational approximation The behavior of the equioscillation points (alternants) for the error in best uniform approximation on [−1, 1] by rational functions of degreen is investigated. In general, the points of the alternants need not be dense in [−1, 1], even when approximation by rational functions of degree (m, n) is considered and asymptoticallym/n ≥ 1. We show, however, that if more thanO(logn) poles of the approximants stay at a positive distance from [−1, 1], then asymptotic denseness holds, at least for a subsequence. Furthermore, we obtain stronger distribution results when λn (0 < λ ≤ 1) poles stay away from [−1, 1]. In the special case when a Markoff function is approximated, the distribution of the equioscillation points is related to the asymptotics for the degree of approximation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

Behavior of alternation points in best rational approximation

Acta Applicandae Mathematicae , Volume 33 (3) – Dec 31, 2004

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References (25)

Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Computational Mathematics and Numerical Analysis; Applications of Mathematics; Partial Differential Equations; Probability Theory and Stochastic Processes; Calculus of Variations and Optimal Control; Optimization
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/BF00995488
Publisher site
See Article on Publisher Site

Abstract

The behavior of the equioscillation points (alternants) for the error in best uniform approximation on [−1, 1] by rational functions of degreen is investigated. In general, the points of the alternants need not be dense in [−1, 1], even when approximation by rational functions of degree (m, n) is considered and asymptoticallym/n ≥ 1. We show, however, that if more thanO(logn) poles of the approximants stay at a positive distance from [−1, 1], then asymptotic denseness holds, at least for a subsequence. Furthermore, we obtain stronger distribution results when λn (0 < λ ≤ 1) poles stay away from [−1, 1]. In the special case when a Markoff function is approximated, the distribution of the equioscillation points is related to the asymptotics for the degree of approximation.

Journal

Acta Applicandae MathematicaeSpringer Journals

Published: Dec 31, 2004

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