Chebyshev polynomials in the complex plane and on the real line

TL;DR

This paper surveys the theory of Chebyshev polynomials, emphasizing their extremal properties and extensions to the complex plane, including new proofs of known theorems and a collection of open problems.

Contribution

It provides a comprehensive overview of Chebyshev polynomials' extremal properties, with new proofs and open problems, enhancing understanding of their complex and real line behaviors.

Findings

Compilation of known theorems with new proofs

Identification of open problems in Chebyshev polynomial theory

Extension of Chebyshev polynomial properties to the complex plane

Abstract

We present a survey of central developments in the theory of Chebyshev polynomials, introduced by P.~L.~Chebyshev and later extended to the complex plane by G.~Faber. Our primary focus is their defining extremal property: among all polynomials with a prescribed leading coefficient, they minimize the supremum norm on a given compact set. Although we do not present new results, we provide -- in selected cases -- new proofs of known theorems and compile a collection of open problems.