Section: New Results

Orthogonal Polynomials

Participant : Laurent Baratchart.

We studied this year the asymptotic behavior of the orthonormal polynomials Pn with respect to a non-negative weight w on a simply connected planar domain Ω:

Ω P n P ¯ k w d m = δ n , k ,

with δn,k the Kronecker symbol. We proved that if Ω has boundary Ω of class C1,α, α>0, and if w converges in some appropriate sense to a boundary function w1Lp(Ω) while not vanishing “too much” at the boundary, then

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

outside the convex hull of Ω, with Φ the conformal map from the complement of Ω onto the complement of the unit disk normalized so that Φ'()=, and Sw1- the so-called exterior Szegő function of w1.

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 ¯ 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.