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Zbl 0801.41021
Braess, Dietrich; Lubinsky, D.S.; Saff, E.B.
Behavior of alternation points in best rational approximation. (English)
[J] Acta Appl. Math. 33, No.2-3, 195-210 (1993). ISSN 0167-8019; ISSN 1572-9036/e

Summary: The behavior of the equioscillation points (alternants) for the error in best uniform approximation on $[-1,1]$ by rational functions of degree $n$ 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 asymptotically $m/n\geq 1$. We show, however, that if more than $O(\log n)$ 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 $\lambda n$ $(0<\lambda \leq 1)$ poles stay away from $[- 1,1]$. In the special case when a Markov function is approximated, the distribution of the equioscillation points is related to the asymptotics for the degree of approximation.

MSC 2000:: *41A50 Best approximation
41A20 Approximation by rational functions

Keywords: best uniform approximation; Markov function

Cited in: Zbl 1057.41008

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