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 $\Omega$ (so-called weighted Bergman polynomials) to the case where $\Omega$ is simply connected, asymptotically conformal and chord arc, with exterior conformal map $f$ from the complement of the disk to the complement of $\Omega$ such that $f^{\text{'}\text{'}}/f^{\text{'}}$ lies in a Hardy class $H^q$ with $q<1$. This class of domain is more general than, say the $C^{1\alpha }$ class. Meanwhile the weight should have integrable non-tangential maximal function and non-tangential limit with positive geometric mean. As $n\to \infty$, the formula reads

locally uniformly outside the convex hull of $\Omega$, where $\Phi =f^{-1}$ and $S_{w\circ f}$ 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 $\Omega$ or else constant or analytic weight. An article is being written on this topic.