Faber Polynomials for m-Fold Symmetric Domains

The Faber polynomials for a region of the complex plane are of interest as a basis for polynomial approximations to analytic functions. In this paper we study the Faber polynomials associated with m -fold symmetric domains. Explicit formulae of Faber polynomials both for symmetric and nonsymmetric lemniscates are derived. In addition, we use a new determinant representation which relates the zeros of Faber polynomials to the eigenvalues of a certain matrix to compute the zeros of Faber polynomials for lemniscates.