Weak-star convergence and a polynomial approximation problem

Abstract

Let $E$ be an arbitrary subset of the unit circle $T$ and let $f$ be a function defined on $E$. When there exist polynomials $P_n$ which are uniformly bounded by a number $M > 0$ on $T$ and converge (pointwise) to $f$ at each point of $E$? We give a necessary and sufficient description of such functions $f$. The necessity part of our result, in fact, is a classical theorem of S.Ya. Khavinson, while the proof of sufficiency uses the method that has been recently applied in particular in the author's solution of an approximation problem proposed by L. Zalcman.