Polynomials of the best uniform approximation to sgn(x) on two intervals

TL;DR

This paper characterizes the best uniform polynomial approximation to the sign function on two intervals using conformal mappings, enabling precise determination of the approximation error's asymptotic behavior.

Contribution

It introduces a novel approach using special conformal mappings to explicitly describe the best uniform approximation to sgn(x) on two intervals.

Findings

Exact asymptotic behavior of the approximation error determined.

Representation of the approximation polynomials via conformal mappings.

Enhanced understanding of polynomial approximation on union of intervals.

Abstract

We describe polynomials of the best uniform approximation to sgn(x) on the union of two intervals in terms of special conformal mappings. This permits us to find the exact asymptotic behavior of the error of this approximation.