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, SzegĹ‘ showed that the zeros of the normalized partial sums, $s_{n}(nz)$, of $e^{z}$ tended to what is now called the *SzegĹ‘ curve* $S$, where \[ S:= \left \{ z \in {\mathbb {C}}:|ze^{1-z}|=1 \text { and } |z| \leq 1 \right \}. \] Using modern methods of weighted potential theory, these zero distribution results of SzegĹ‘ 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 \[ \lim _{n \rightarrow \infty } \|e^{-nz} P_{n}(z)-f(z)\|_{\overline {G}} =0, \] where $\| \cdot \|_{\overline {G}}$ denotes the supremum norm on $\overline {G}$. 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

MR Author ID:
319712

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

Keywords:
SzegĹ‘ 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