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Russian Mathematical Surveys, 2021, Volume 76, Issue 3, Pages 379–387
DOI: https://doi.org/10.1070/RM9991 (Mi rm9991) 		 
Convergence of Bieberbach polynomials: Keldysh's theorems and Mergelyan's conjecture

A. I. Aptekarev

Keldysh Institute of Applied Mathematics of Russian Academy of Sciences
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DOI: https://doi.org/10.1070/RM9991
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.
Received: 27.12.2020
Russian version:
Uspekhi Matematicheskikh Nauk, 2021, Volume 76, Issue 3(459), Pages 3–12
DOI: https://doi.org/10.4213/rm9991
Bibliographic databases:
Document Type: Article
UDC: 517.53
MSC: 01A70, 30B60, 30C35
Language: English
Original paper language: Russian
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
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