Properties and examples of Faber--Walsh polynomials

The Faber--Walsh polynomials are a direct generalization of the (classical) Faber polynomials from simply connected sets to sets with several simply connected components. In this paper we derive new properties of the Faber--Walsh polynomials, where we focus on results of interest in numerical linear algebra, and on the relation between the Faber--Walsh polynomials and the classical Faber and Chebyshev polynomials. Moreover, we present examples of Faber--Walsh polynomials for two real intervals as well as some non-real sets consisting of several simply connected components.