On a polynomial approximation problem

AbstractLet F be a closed subset of the unit circle T and let f∈C(F). We investigate the problem of uniform approximation of f on F by polynomials Pn which are uniformly bounded on the unit disk Δ. In a particular case when F is a closed arc of T, 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 f given by Zalcman for the case of an arc also holds in the general case