Strong asymptotics for Bergman polynomials over non-smooth domains

Stylianopoulos, Nikos

doi:10.1016/j.crma.2009.11.007

Complex Analysis/Numerical Analysis

Strong asymptotics for Bergman polynomials over non-smooth domains

Presented by: Philippe G. Ciarlet

Stylianopoulos, Nikos ¹

¹ Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus

Comptes Rendus. Mathématique, Volume 348 (2010) no. 1-2, pp. 21-24

Abstract (Original language)
Résumé

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

p_{n} (z) = λ_{n} z^{n} + \dots, λ_{n} > 0, n = 0, 1, 2, \dots,

of polynomials that are orthonormal with respect to the inner product

〈 f, g 〉 : = \int_{G} f (z) \bar{g (z)} d A (z)

, where dA stands for the area measure. The aim of this Note is to report on results regarding the strong asymptotics of

p_{n}

and

λ_{n}

,

n \in N

, under the assumption that Γ is piecewise analytic. These results complement an investigation started in 1923 by T. Carleman, who derived the strong asymptotics for domains with analytic boundaries and carried over by P.K. Suetin in the 1960's, who established them for domains with smooth boundaries.

Soit G un domaine simplement connexe dans le plan complexe $C$ , avec une frontière $Γ : = \partial G$ qui est une courbe de Jordan, et soit ${p_{n}}_{n = 0}^{\infty}$ les polynômes de Bergman associés a G. Plus precisémént la suite

p_{n} (z) = λ_{n} z^{n} + \dots, λ_{n} > 0, n = 0, 1, 2, \dots,

des polynômes de Bergman est orthonormal pour le produit scalaire

〈 f, g 〉 : = \int_{G} f (z) \bar{g (z)} d A (z)

, ou dA est la mesure de surface. On obtient des estimations asymptotiques fortes pour

p_{n}

et

λ_{n}

,

n \in N

, sous l'hypothèse que Γ est analytique par morceaux. Le resultat obtenu complète une étude commencés par T. Carleman en 1923, pour des domaines avec une frontière analytique, et continué par P.K. Suetin dans les années 1960, pour des domaines avec une frontiere régulière.

Received: 2009-10-06
Accepted: 2009-11-09
Published online: 2009-12-24

DOI: 10.1016/j.crma.2009.11.007

Author's affiliations:

Stylianopoulos, Nikos ¹

¹ Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus

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Stylianopoulos, Nikos. Strong asymptotics for Bergman polynomials over non-smooth domains. Comptes Rendus. Mathématique, Volume 348 (2010) no. 1-2, pp. 21-24. doi: 10.1016/j.crma.2009.11.007

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Cited by

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