Zolotarev's fifth and sixth problems

TL;DR

This paper explores optimal rational approximants to functions like sqrt(z) and sign(z) on the unit circle, linking solutions to classical approximation problems posed by Zolotarev in 1877.

Contribution

It introduces and solves two new approximation problems, connecting them to Zolotarev's earlier work in a novel way.

Findings

Identifies best rational approximants with modulus 1 on the unit circle.

Establishes a nontrivial relationship between new and classical Zolotarev problems.

Extends understanding of rational approximation near discontinuities.

Abstract

In an influential 1877 paper, Zolotarev asked and answered four questions about polynomial and rational approximation. We ask and answer two questions: what are the best rational approximants $r$ and $s$ to $\sqrt{z}$ and $\mbox{sign}(z)$ on the unit circle (excluding certain arcs near the discontinuities), with the property that $|r(z)|=|s(z)|=1$ for $|z|=1$? We show that the solutions to these problems are related to Zolotarev's third and fourth problems in a nontrivial manner.