Computation of Zolotarev rational functions

TL;DR

This paper introduces an algorithm for computing Zolotarev rational functions, which are optimal in minimizing their size on one set relative to another, and also approximates the sign function in this context.

Contribution

The paper presents a novel algorithm for computing Zolotarev rational functions and extends the approach to approximate the sign function for complex sets.

Findings

Algorithm successfully computes Zolotarev rational functions.

Extension to sign function approximation demonstrated.

Provides a practical method for optimal rational approximation in complex domains.

Abstract

An algorithm is presented to compute Zolotarev rational functions, that is, rational functions $r_n^*$ of a given degree that are as small as possible on one set $E\subseteq\complex\cup\{\infty\}$ relative to their size on another set $F\subseteq\complex\cup\{\infty\}$ (the third Zolotarev problem). Along the way we also approximate the sign function relative to $E$ and $F$ (the fourth Zolotarev problem).