Abstract
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 ≥ 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 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.
The research of this author was supported, in part, by NSF grant DMS 920–3659.
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Braess, D., Lubinsky, D.S., Saff, E.B. (1994). Behavior of Alternation Points in Best Rational Approximation. In: Cuyt, A. (eds) Nonlinear Numerical Methods and Rational Approximation II. Mathematics and Its Applications, vol 296. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0970-3_13
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DOI: https://doi.org/10.1007/978-94-011-0970-3_13
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