Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-97-01889-8
MathSciNet review: 1407500
Full-text PDF

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.


References [Enhancements On Off] (What's this?)

  • 1. P. B. Borwein and W. Chen, Incomplete rational approximation in the complex plane, Constr. Approx. 11 (1995), 85-106. MR 95k:41024
  • 2. J. D. Buckholtz, A characterization of the exponential series, Amer. Math. Monthly 73, Part II (1966), 121-123. MR 34:2838
  • 3. R. S. Varga and A. J. Carpenter, Asymptotics for the zeros of the partial sums of $e^{z}$.II, Computational Methods and Function Theory, Lecture Notes in Math., vol. 1435, pp. 201-207, Springer-Verlag, Heidelberg, 1990. MR 92m:33004
  • 4. A. J. Carpenter, R. S. Varga and J. Waldvogel, Asymptotics for the zeros of the partial sums of $e^{z}$. I., Rocky Mount. J. of Math. 21 (1991), 99-119. MR 92m:33003
  • 5. D. Gaier, Lectures on Complex Approximation, Birkhäuser, Boston, 1987. MR 88i:30059b
  • 6. P. Henrici, Applied and Computational Complex Analysis, vol. 2, John Wiley and Sons, New York, 1977. MR 56:12235
  • 7. N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 1972. MR 50:2520
  • 8. G. G. Lorentz, Approximation by incomplete polynomials (problems and results), Padé and Rational Approximations: Theory and Applications (E. B. Saff and R. S. Varga, eds.), pp. 289-302, Academic Press, New York, 1977. MR 57:6956
  • 9. H. N. Mhaskar and E. B. Saff, The distribution of zeros of asymptotically extremal polynomials, J. Approx. Theory 65 (1991), 279-300. MR 92d:30005
  • 10. H. N. Mhaskar and E. B. Saff, Weighted analogues of capacity, transfinite diameter and Chebyshev constant, Constr. Approx. 8 (1992), 105-124. MR 93a:31004
  • 11. E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, Heidelberg, 1997.
  • 12. G. Szeg\H{o}, Über eine Eigenshaft der Exponentialreihe, Sitzungsber. Berl. Math. Ges. 23 (1924), 50-64.
  • 13. V. Totik, Weighted Approximation with Varying Weight, Lecture Notes in Math., vol. 1569, Springer-Verlag, Heidelberg, 1994. MR 96f:41002
  • 14. J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, Colloquium Publications, vol. 20, Amer. Math. Soc., Providence, 1969. MR 36:1672b (earlier ed.)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 30E10, 30C15, 31A15, 41A30

Retrieve articles in all journals with MSC (1991): 30E10, 30C15, 31A15, 41A30


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

American Mathematical Society