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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.