Behavior of alternation points in best rational approximation

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.