Circularity of the Error Curve and Sharpness of the CF Method in Complex Chebyshev Approximation

Abstract

Let $f(z)$ be analytic at the origin, and for $\varepsilon > 0$, let $f(\varepsilon z)$ be best approximated in the Chebyshev sense on the unit disk by a rational function of type $(m,n)$. It has been shown previously by the CF method that the error curve for this approximation deviates from a circle by at most $O(\varepsilon ^{2m + 2n + 3} )$ as $\varepsilon \to 0$. We prove here that this bound is sharp in two senses: the error curve for a given function cannot be asymptotically more circular than the CF method predicts; moreover there exist functions for which the near-circularity is of order$\varepsilon ^{2m + 2n + 3} $ but no smaller.