Section: New Results

Approximation

Participant : Laurent Baratchart.

Orthogonal Polynomials

This is joint work with Nikos Stylianopoulos (Univ. of Cyprus).

We study the asymptotic behavior of weighted orthogonal polynomials on a bounded simply connected plane domain $\Omega$. The $n$-th orthogonal polynomial $P_n$ has degree $n$, positive leading coefficient, and satisfies

where $w$ is an integrable positive weight and ${\delta }_{n,k}$ is the Kronecker symbol. When $\Omega$ is smooth while $w$ is Hölder-continuous and non-vanishing, it is known that

locally uniformly outside the convex hull of $\overline{\Omega }$, where $\Phi$ is the conformal map from the complement of $\Omega$ onto the complement of the unit disk and $S_w$ is the so-called Szegö function of the trace of $w$ on the boundary $\partial \Omega$ [81] . If we compare it with classical exterior Szegő asymptotics, the formula asserts that $P_n$ behaves asymptotically like the $n$-th orthogonal polynomial with respect to a weight supported on $\partial \Omega$ (the trace of $w$), up to a factor $\sqrt{(n+1)/\pi }$.

When $\Omega$ is the unit disk, we proved this result under unprecedented weak assumptions on $w$, namely $w\left(r{e}^{i\theta }\right)$ should converge in $L^p\left(T\right)$ as $r\to 1$ for some $p>1$ and its ${log}^{-}$ should be bounded in the real Hardy space $H^1$. An article is being written on these findings and the case of a smooth domain $\Omega$, more general than a disk, is under examination.

Meromorphic approximation

This is joint work with Maxim Yattselev (Purdue Univ. at Indianapolis, USA).

We proved in [6] that the normalized counting measure of poles of best $H^2$ approximants of degree $n$ to a function analytically continuable, except over finitely many branchpoints lying outside the unit disk, converges to the Green equilibrium distribution of the compact set of minimal Green capacity outside of which the function is single valued (the normalized counting measure is the probability measure with equal mass at each pole). This result warrants source recovery techniques used in section 6.1.1 . Here we consider the corresponding problem for best uniform meromorphic approximants on the unit circle (so-called AAK approximants after Adamjan, Arov and Krein), in the case where the function may have poles and essential singularities. This year, we established a similar result when the function has finitely many essential singularities. The general case is still pending.