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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/n1. We show, however, that if more than O(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<λ1) 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:
41A50Best approximation, Chebyshev systems
41A20Approximation by rational functions
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