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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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
MathSciNet review: 1407500
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Abstract | References | Similar Articles | Additional Information

Abstract: 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

\begin{displaymath}S:= \left \{ z \in {\mathbb {C}}:|ze^{1-z}|=1 \text { and } |z| \leq 1 \right \}. \end{displaymath}

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

\begin{displaymath}\displaystyle \lim_{n \rightarrow \infty } \|e^{-nz} P_{n}(z)-f(z)\|_{\overline {G}} =0, \end{displaymath}

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
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: http://dx.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