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

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.