Chebyshev polynomials in the complex plane and on the real line

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.

Major updates to the entire document. Section 2.2 - proof of orthogonality properties of Chebyshev polynomials, relation to extremal signatures. Section 3.1 - new proof of Suetins asymptotics for Faber polynomials. Section 3.3 is shortened. Section 3.5 includes details Chebyshev polynomials on equipotential curves