On a polynomial approximation problem

Let FF be a closed subset of the unit circle TT and let f∈C(F)f∈C(F). We investigate the problem of uniform approximation of ff on FF by polynomials PnPn which are uniformly bounded on the unit disk ΔΔ. In a particular case when FF is a closed arc of TT, the problem was solved by L. Zalcman in 1982, who has also pointed out the possibility of considering more general approximation sets instead of an arc. The present paper gives a necessary and sufficient solution of the above problem. In fact we show that the (simple) description of ff given by Zalcman for the case of an arc also holds in the general case.