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Uspekhi Mat. Nauk, 2021, Volume 76, Issue 3(459), Pages 3–12 (Mi umn9991)  

Convergence of Bieberbach polynomials: Keldysh's theorems and Mergelyan's conjecture

A. I. Aptekarev

Keldysh Institute of Applied Mathematics of Russian Academy of Sciences

Abstract: Results due to Keldysh on the convergence of Bieberbach polynomials and the density of polynomials in spaces of analytic functions are considered. Their further development and relevance in the contemporary context of constructive complex analysis are discussed. Particular focus is placed on Mergelyan's conjecture on the rate of convergence in a domain with smooth boundary, which is still open.
Bibliography: 20 titles.

Keywords: Bieberbach polynomials; extremal properties of analytic functions; approximate conformal mappings; completeness of polynomials; orthogonal polynomials with respect to the area.

DOI: https://doi.org/10.4213/rm9991

Full text: PDF file (686 kB)
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English version:
Russian Mathematical Surveys, 2021, 76:3, 379–387

Bibliographic databases:

UDC: 517.53
MSC: 01A70, 30B60, 30C35
Received: 27.12.2020

Citation: A. I. Aptekarev, “Convergence of Bieberbach polynomials: Keldysh's theorems and Mergelyan's conjecture”, Uspekhi Mat. Nauk, 76:3(459) (2021), 3–12; Russian Math. Surveys, 76:3 (2021), 379–387

Citation in format AMSBIB
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