Speaker: Erwin Miña Díaz (University of Mississippi)
Title: Asymptotics of Bergman polynomials for domains with reflection-invariant corners
Abstract: In the talk, I will present new results on the strong asymptotic behavior of polynomials $(p_n){n=0}^{\infty}$ that are orthogonal over a domain $D$ with piecewise analytic boundary. More specifically, $D$ is assumed to have the property that conformal maps $\vfi$ of $D$ onto the unit disk extend analytically across the boundary $L$ of $D$, and that $\vfi'$ has a finite number of zeros $z_1, \ldots,z_q$ on $L$. The boundary $L$ is then piecewise analytic with corners at the zeros of $\vfi'$. A result of Stylianopoulos implies that a Carleman-type strong asymptotic formula for $p_n$ holds on the exterior domain $\C\setminus\overline{D}$. We prove that the same formula remains valid across $L\setminus\{z_1, \ldots,z_q\}$ 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 $\vfi$ 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. The results were obtained in collaboration with Aron Wennman from KU Leuven.
Place: https://ucf.zoom.us/j/96789802247
Meeting ID: 967 8980 2247
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