Asymptotics and zeros of Bergman polynomials, Erwin Miña Díaz
Автор: Ibero-American Seminar
Загружено: 2025-06-30
Просмотров: 47
[20/06/2025] Asymptotics and zeros of Bergman polynomials for domains with reflection-invariant corners, Erwin Miña Díaz, Mathematics Department at the University of Mississippi, MS, USA.
Abstract: I will present recent results on the strong asymptotic behavior and limiting zero distribution 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 $\varphi$ of $D$ onto the unit disk extend analytically across the boundary $L$ of $D$, and that $\varphi'$ 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 $\varphi'$. We prove that a Carleman-type strong asymptotic formula for $p_n$ holds outside a certain compact set $K$ that contains each corner of $L$ but otherwise sits entirely inside $D$. Near each corner, $K$ consists of an analytic arc departing from the corner. As $n\to \infty$, the zeros of $p_n$ accumulate on $K$ and every boundary point of $K$ is a zero limit point.
This is joint work with Aron Wennman.
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