On the Distribution of Zeros and Poles of Rational Approximants on Intervals

Abstract

The distribution of zeros and poles of best rational approximants is well understood for the functions 𝑓(𝑥)=|𝑥|𝛼, 𝛼>0. If 𝑓∈𝐶[−1,1] is not holomorphic on [−1,1], the distribution of the zeros of best rational approximants is governed by the equilibrium measure of [−1,1] under the additional assumption that the rational approximants are restricted to a bounded degree of the denominator. This phenomenon was discovered first for polynomial approximation. In this paper, we investigate the asymptotic distribution of zeros, respectively, 𝑎-values, and poles of best real rational approximants of degree at most 𝑛 to a function 𝑓∈𝐶[−1,1] that is real-valued, but not holomorphic on [−1,1]. Generalizations to the lower half of the Walsh table are indicated.