Strong Asymptotics for Bergman Polynomials over Domains with Corners and Applications

Abstract

Let G be a bounded simply-connected domain in the complex plane ℂ, whose boundary Γ:=∂G is a Jordan curve, and let \(\{p_{n}\}_{n=0}^{\infty}\) denote the sequence of Bergman polynomials of G. This is defined as the unique sequence

$$p_n(z) = \lambda_n z^n+\cdots, \quad \lambda_n>0,\ n=0,1,2,\ldots, $$

of polynomials that are orthonormal with respect to the inner product

$$\langle f,g\rangle := \int_G f(z) \overline{g(z)} \,dA(z), $$

where dA stands for the area measure.

We establish the strong asymptotics for p n and λ n , n∈ℕ, under the assumption that Γ is piecewise analytic. This complements an investigation started in 1923 by T. Carleman, who derived the strong asymptotics for Γ analytic, and carried over by P.K. Suetin in the 1960s, who established them for smooth Γ. In order to do so, we use a new approach based on tools from quasiconformal mapping theory. The impact of the resulting theory is demonstrated in a number of applications, varying from coefficient estimates in the well-known class Σ of univalent functions and a connection with operator theory, to the computation of capacities and a reconstruction algorithm from moments.

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Correspondence to Nikos Stylianopoulos.

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Communicated by Edward B. Saff.

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Stylianopoulos, N. Strong Asymptotics for Bergman Polynomials over Domains with Corners and Applications. Constr Approx 38, 59–100 (2013). https://doi.org/10.1007/s00365-012-9174-y

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Keywords

  • Bergman orthogonal polynomials
  • Faber polynomials
  • Strong asymptotics
  • Polynomial estimates
  • Quasiconformal mapping
  • Conformal mapping

Mathematics Subject Classification (2010)

  • 30C10
  • 30C30
  • 30C50
  • 30C62
  • 41A10
  • 65E05
  • 30E05