The Bernstein Problem in Two Dimensions

I will outline a complete solution to the weighted approximation problem for polynomials

on an arbitrary bounded simply connected domain  in the complex plane. In that setting

the problem was first studied extensively by Keldysh prior to 1941 in the context of L2-

approximation, and more than four decades later by Beurling where, in the latter instance,

the emphasis was on uniform approximation. Here, Beurling obtains the sharper result

with respect to the weight w, but at the expense of limiting the type of region to which

his argument applies. Ironically, however, the two problems turned out to be essentially

equivalent, but neither Beurling nor Keldysh obtained what might be considered a definitive

solution. My presentation will focus on the L2-case, where the theory of Sobolev spaces and

its associated potential theory is available. It is a simple matter to pass from there to a

solution in the case of uniform approximation.