Asymptotics of Bergman polynomials for domains with reflection-invariant corners

Abstract

We study the asymptotic behavior of the Bergman orthogonal polynomials $(p_n{)}_{n=0}^{\infty }$ for a class of bounded simply connected domains $D$. The class is defined by the requirement that conformal maps $\phi$ of $D$ onto the unit disk extend analytically across the boundary $L$ of $D$, and that ${\phi }^{\prime }$ has a finite number of zeros $z_1,\dots ,z_q$ on $L$. The boundary $L$ is then piecewise analytic with corners at the zeros of ${\phi }^{\prime }$. A result of Stylianopoulos implies that a Carleman-type strong asymptotic formula for $p_n$ holds on the exterior domain $\mathbb{C}\setminus \overline{D}$. We prove that the same formula remains valid across $L\setminus \{z_1,\dots ,z_q\}$ and on a maximal open subset of $D$. As a consequence, the only boundary points that attract zeros of $p_n$ are the corners. This is in stark contrast to the case when $\phi$ fails to admit an analytic extension past $L$, since when this happens the zero counting measure of $p_n$ is known to approach the equilibrium measure for $L$ along suitable subsequences.

36 pages, 3 figures