## Section: New Results

### Orthogonal Polynomials

Participant : Laurent Baratchart.

We studied this year the asymptotic behavior of the orthonormal polynomials ${P}_{n}$ with respect to a non-negative weight $w$ on a simply connected planar domain $\Omega $:

with ${\delta}_{n,k}$ the Kronecker symbol. We proved that if $\Omega $ has boundary $\partial \Omega $ of class ${C}^{1,\alpha}$, $\alpha >0$, and if $w$ converges in some appropriate sense to a boundary function ${w}_{1}\in {L}^{p}\left(\partial \Omega \right)$ while not vanishing “too much” at the boundary, then

outside the convex hull of $\Omega $, with $\Phi $ the conformal map from the complement of $\Omega $ onto the complement of the unit disk normalized so that ${\Phi}^{\text{'}}\left(\infty \right)=\infty $, and ${S}_{{w}_{1}}^{-}$ the so-called exterior Szegő function of ${w}_{1}$.

This generalizes considerably known asymptotics on
analytic domains with Hölder smooth non vanishing weights [10] .
The proof rests on some Hardy space theory, conformal mapping and
$\overline{\partial}$ techniques. An exposition of the result was given at
the conference
*Orthogonal and Multiple Orthogonal Polynomials*, August 9-14 2015, Oaxaca
(Mexico). An article is being written to report on this result.