THE SZEGO CURVE, ZERO DISTRIBUTION AND WEIGHTED APPROXIMATION

In 1924, Szego showed that the zeros of the normalized partial sums, s (n)(nz), of e(z) tended to what is now called the Szego curve S, where S := {z is an element of C : \ze(1-z)\ = 1 and \z\ less than or equal to 1}. Using modern methods of weighted potential theory, these zero distribution results of Szego can be essentially recovered, along with an asymptotic formula for the weighted partial sums {e(-nz)s(n)(nz)}( n=0)(infinity). We show that G := Int S is the largest universal domai n such that the weighted polynomials e(-nz)P(n)(z) are dense in the se t of functions analytic in G. As an example of such results, it is sho wn that if f(z) is analytic in G and continuous on (G) over bar with f (1) = 0, then there is a sequence of polynomials {P-n(z)}(n=0)(infinit y), with deg P-n less than or equal to n, such that Lim(n-->infinity) \\e(-nz)P(n)(z)-f(z)\\(G)-=0, where \\.\\(G)- denotes the supremum nor m of (G) over bar. Similar results are also derived for disks.