Asymptotics of polynomials orthogonal over circular multiply connected domains

Abstract

Let $D$ be a domain obtained by removing, out of the unit disk $\{z:|z|<1\}$, finitely many mutually disjoint closed disks, and for each integer $n\ge 0$, let $P_n(z)=z^n+\cdots$ be the monic $n$th-degree polynomial satisfying the planar orthogonality condition ${\int }_D{P}_n(z)\overline{z^m}dxdy=0$, $0\le m<n$. Under a certain assumption on the domain $D$, we establish asymptotic expansions and formulae that describe the behavior of $P_n(z)$ as $n\to \infty$ at every point $z$ of the complex plane. We also give an asymptotic expansion for the squared norm ${\int }_D|P_n{|}^{2}dxdy$.

References in corpus (3)

• Asymptotic Properties of Extremal Polynomials Corresponding to Measures Supported on Analytic Regions
• An asymptotic integral representation for Carleman orthogonal polynomials
• Asymptotics of Carleman polynomials for level curves of the inverse of a shifted Zhukovsky transformation