On the Zeros of Weighted Faber Polynomials

Let E be a compact set in the complex plane C containing more than one point and having simply connected complement in the extended complex plane C¯¯¯¯. Denote by Fn(z;g) the Faber polynomial of degree n for E with weight function g(z). It is shown that if g(z) is an analytic function on C¯¯¯¯∖E with g(∞)>0 and singularity on ∂E then every point of ∂E is a limit point of the set of zeros of{Fn(z;g)}. To illustrate this result the author computes the zeros of the weighted Faber polynomials for an m-cusped hypocycloid (see also a paper by M. X. He and E. B. Saff [J. Approx. Theory 78 (1994), no. 3, 410–432; MR1292970] and the references therein).