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
of polynomials that are orthonormal with respect to the inner product
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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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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DOI: https://doi.org/10.1007/s00365-012-9174-y
Keywords
- Bergman orthogonal polynomials
- Faber polynomials
- Strong asymptotics
- Polynomial estimates
- Quasiconformal mapping
- Conformal mapping