Strong asymptotics for Bergman polynomials over non-smooth domains
Date
2010ISSN
1631-073XSource
Comptes Rendus MathematiqueVolume
348Issue
1-2Pages
21-24Google Scholar check
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Let G be a bounded simply-connected domain in the complex plane C, whose boundary Γ : = ∂ G is a Jordan curve, and let {pn}n = 0 ∞ denote the sequence of Bergman polynomials of G. This is defined as the sequencepn (z) = λn zn + ⋯, λn > 0, n = 0, 1, 2, ..., of polynomials that are orthonormal with respect to the inner product 〈 f, g 〉 : = ∫G f (z) over(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 pn and λn, n ∈ 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. © 2009 Académie des sciences.