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 (so-called weighted Bergman polynomials) to the case where is simply connected, asymptotically conformal and chord arc, with exterior conformal map from the complement of the disk to the complement of such that lies in a Hardy class with . This class of domain is more general than, say the class. Meanwhile the weight should have integrable non-tangential maximal function and non-tangential limit with positive geometric mean. As , the formula reads
locally uniformly outside the convex hull of , where and 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 or else constant or analytic weight. An article is being written on this topic.