Some properties of Faber-Walsh polynomials

TL;DR

This paper explores properties of Faber-Walsh polynomials, including their representations, asymptotic behavior, and optimality, extending classical Faber polynomial theory to more general compact sets.

Contribution

It introduces equivalent representations and asymptotic properties of Faber-Walsh polynomials, demonstrating their asymptotic optimality for certain compact sets.

Findings

Derived equivalent representations of Faber-Walsh polynomials

Established asymptotic properties on the complement of the compact set

Showed normalized Faber-Walsh polynomials are asymptotically optimal

Abstract

Walsh introduced a generalisation of Faber polynomials to certain compact sets which need not be connected. We derive several equivalent representations of these Faber-Walsh polynomials, analogous to representations of Faber polynomials. Some simple asymptotic properties of the Faber-Walsh polynomials on the complement of the compact set are established. We further show that suitably normalised Faber-Walsh polynomials are asymptotically optimal polynomials in the sense of [Eiermann and Niethammer 1983].