Section: New Results

Asymptotics of weighted Bergman polynomials

Participant : Laurent Baratchart.

We extended this year exterior asymptotics for orthonormal polynomials with respect to a weight on a planar region Ω (so-called weighted Bergman polynomials) to the case where Ω is simply connected, asymptotically conformal and chord arc, with exterior conformal map f from the complement of the disk to the complement of Ω such that f''/f' lies in a Hardy class Hq with q<1. This class of domain is more general than, say the C1α class. Meanwhile the weight should have integrable non-tangential maximal function and non-tangential limit with positive geometric mean. As n, the formula reads

P n ( z ) = n + 1 π 1 / 2 S w f ( Φ ( z ) ) Φ n ( z ) Φ ' ( z ) { 1 + o ( 1 ) } ,

locally uniformly outside the convex hull of Ω, where Φ=f-1 and Swf is the Szegő function of the boundary weight . The proof uses quasi-conformal mappings and some Hardy space theory, along with classical Fourier analysis of Taylor sections.

The result goes much beyond those previously known, which either assume analyticity of Ω or else constant or analytic weight. An article is being written on this topic.