Approximation of sgn(x) on two symmetric intervals by rational functions with fixed poles

TL;DR

This paper extends the explicit solutions for best polynomial approximation of sgn(x) over two intervals to odd rational functions with fixed poles, using conformal mappings and convexity of comb domains.

Contribution

It introduces a new approach for rational approximation of sgn(x) on symmetric intervals, generalizing previous polynomial approximation results with explicit solutions.

Findings

Explicit solutions for best rational approximation of sgn(x) with fixed poles.

Use of conformal mappings and convexity properties to prove existence.

Extension of polynomial approximation techniques to rational functions.

Abstract

Recently A. Eremenko and P. Yuditskii found explicit solutions of the best polynomial approximation problems of sgn(x) over two intervals in terms of conformal mappings onto special comb domains. We give analogous solutions for the best approximation problems of sgn(x) over two symmetric intervals by odd rational functions with fixed poles. Here the existence of the related conformal mapping is proved by using convexity of the comb domains along the imaginary axis.