On a problem of Beurling

This article is an extended version of a lecture presented to a conference at Brown University on 11 June 2017 in celebration of John Wermer's 90th birthday. Here, we discuss a complete solution to the weighted approximation problem for polynomials on an arbitrary bounded simply connected domain $\Omega$ in the complex plane. The problem had been studied extensively by Keldysh [13] prior to 1941 in the context of weighted $L^2$-approximation, and more recently by Beurling [5], where the emphasis is on uniform approximation.