Faber Polynomials on Quaternionic Compact Sets

Abstract

We introduce the concept of Faber polynomials attached to an axially symmetric compact set \(K\subset \mathbb {H}\) with \(\overline{\mathbb {H}}{\setminus } K\) simply connected and we obtain several results on these polynomials in the quaternionic setting, including expansions in Faber series of functions continuous on K and slice regular in the interior of K. In this paper, by quaternionic polynomials we mean polynomials with quaternionic coefficients written on the right. The restriction of quaternionic Faber polynomials and series expansions to axially symmetric sets is not reductive. On the contrary, it is naturally imposed by two facts: the first is that the quaternionic Riemann mapping theorem does not hold for general compact sets but only for the axially symmetric ones; the second is that the natural domains of definition of the slice regular functions are axially symmetric. The cases of some concrete particular sets are also described with more details.