Weak-star convergence and a polynomial approximation problem

TL;DR

This paper characterizes functions on subsets of the unit circle that can be pointwise approximated by uniformly bounded polynomials, extending classical results using modern approximation methods.

Contribution

It provides a necessary and sufficient condition for polynomial approximation of functions on the unit circle with bounded polynomials, combining classical and recent techniques.

Findings

Characterizes functions approximable by bounded polynomials on the unit circle.

Extends classical approximation theorems with new sufficiency proofs.

Uses modern methods inspired by Zalcman's 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.