On the Behavior of Zeros and Poles of Best Uniform Polynomial and Rational Approximants

Abstract

We investigate the behavior of zeros of best uniform polynomial approximants to a function f, which is continuous in a compact set E ⊂ ℂ and analytic on E, but not on E. Our results are related to a recent theorem of Blatt, Saff, and Simkani which roughly states that the zeros of a subsequence of best polynomial approximants distribute like the equilibrium measure for E. In contrast, we show that there might be another subsequence with zeros essentially all tending to ∞. Also, we investigate near best approximants. For rational best approximants we prove that its zeros and poles cannot all stay outside a neighborhood of E, unless f is analytic on E.