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

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.