Properties and examples of Faber--Walsh polynomials

TL;DR

This paper explores properties of Faber--Walsh polynomials, extending classical Faber polynomials to multi-component sets, with applications in numerical linear algebra and examples for real and complex sets.

Contribution

It introduces new properties of Faber--Walsh polynomials, highlighting their relation to classical polynomials and providing explicit examples for complex sets.

Findings

New properties of Faber--Walsh polynomials derived

Relations established with classical Faber and Chebyshev polynomials

Examples provided for real and non-real sets

Abstract

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.