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.

**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 .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*. 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.)**

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