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Behavior of alternation points in best rational approximation. (English) Zbl 0801.41021
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.

41A50Best approximation, Chebyshev systems
41A20Approximation by rational functions
Full Text: DOI
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