The Szego curve, zero distribution and weighted approximation

In 1924, Szeg\H{o} showed that the zeros of the normalized partial sums, $s_{n}(nz)$, of $e^{z}$ tended to what is now called the Szeg\H{o} curve $S$, where

Using modern methods of weighted potential theory, these zero distribution results of Szeg\H{o} can be essentially recovered, along with an asymptotic formula for the weighted partial sums $\{e^{-nz}s_{n} (nz)\}^{\infty }_{n=0}$. We show that $G:= {\operatorname {Int}} S$ is the largest universal domain such that the weighted polynomials $e^{-nz} P_{n}(z)$ 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 $\overline {G}$ with $f(1)=0$, then there is a sequence of polynomials $\left \{P_{n}(z)\right \}^{\infty }_{n=0}$, with $\deg P_{n} \leq n$, such that

where $\| \cdot \|_{\overline {G}}$ denotes the supremum norm on $\overline {G}$. Similar results are also derived for disks.