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 $f(x)={|x|}^{\alpha }$, $\alpha >0$. If $f\in C[-1,1]$ 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 $f\in C[-1,1]$ that is real-valued, but not holomorphic on $[-1,1]$. Generalizations to the lower half of the Walsh table are indicated.