The Szego curve, zero distribution

and weighted approximation

Authors:
Igor E. Pritsker and Richard S. Varga

Journal:
Trans. Amer. Math. Soc. **349** (1997), 4085-4105

MSC (1991):
Primary 30E10; Secondary 30C15, 31A15, 41A30

DOI:
https://doi.org/10.1090/S0002-9947-97-01889-8

MathSciNet review:
1407500

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In 1924, Szeg\H{o} showed that the zeros of the normalized partial sums, , of tended to what is now called the *Szeg\H{o} curve* , 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 . We show that is the largest universal domain such that the weighted polynomials are dense in the set of functions analytic in . As an example of such results, it is shown that if is analytic in and continuous on with , then there is a sequence of polynomials , with , such that

where denotes the supremum norm on . Similar results are also derived for disks.

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Additional Information

**Igor E. Pritsker**

Affiliation:
Institute for Computational Mathematics, Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242

Email:
pritsker@mcs.kent.edu

**Richard S. Varga**

Affiliation:
Institute for Computational Mathematics, Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242

Email:
varga@mcs.kent.edu

DOI:
https://doi.org/10.1090/S0002-9947-97-01889-8

Keywords:
Szeg\H{o} curve,
weighted polynomials,
weighted energy problem,
extremal measure,
logarithmic potential,
balayage,
modified Robin constant

Received by editor(s):
March 30, 1996

Article copyright:
© Copyright 1997
American Mathematical Society