Asymptotics of Bergman polynomials for domains with reflection-invariant corners
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arXiv
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| Natura: | Preprint |
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2024
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| author | Miña-Díaz, Erwin Wennman, Aron |
| author_facet | Miña-Díaz, Erwin Wennman, Aron |
| contents | 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 $φ$ of $D$ onto the unit disk extend analytically across the boundary $L$ of $D$, and that $φ'$ 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 $φ'$. 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 $L\setminus\{z_1,\ldots,z_q\}$ 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 $φ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09335 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotics of Bergman polynomials for domains with reflection-invariant corners Miña-Díaz, Erwin Wennman, Aron Complex Variables Classical Analysis and ODEs 42C05, 30C10 (Primary) 30E15, 30E20 (Secondary) 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 $φ$ of $D$ onto the unit disk extend analytically across the boundary $L$ of $D$, and that $φ'$ 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 $φ'$. 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 $L\setminus\{z_1,\ldots,z_q\}$ 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 $φ$ 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. |
| title | Asymptotics of Bergman polynomials for domains with reflection-invariant corners |
| topic | Complex Variables Classical Analysis and ODEs 42C05, 30C10 (Primary) 30E15, 30E20 (Secondary) |
| url | https://arxiv.org/abs/2404.09335 |