The Szegő Curve, Zero Distribution and Weighted Approximation

In 1924, Szegő showed that the zeros of the normalized partial sums, sn(nz), of ez tended to what is now called the Szegő curve S, where S := { z ∈ C : |ze1−z| = 1 and |z| ≤ 1 } . Using modern methods of weighted potential theory, these zero distribution results of Szegő can be essentially recovered, along with an asymptotic formula for the weighted partial sums {esn(nz)}n=0. We show that G := Int S is the largest universal domain such that the weighted polynomials e−nzPn(z) are dense in the set of functions analytic in G. As an example of such results, it is shown that if f(z) is analytic in G and continuous on G with f(1) = 0, then there is a sequence of polynomials {Pn(z)}n=0, with degPn ≤ n, such that lim n→∞ ‖e−nzPn(z)− f(z)‖G = 0, where ‖·‖G denotes the supremum norm on G. Similar results are also derived for disks.