Chebyshev polynomials on equipotential curves
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908352313819136 |
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| author | Miña-Díaz, Erwin Rubin, Olof |
| author_facet | Miña-Díaz, Erwin Rubin, Olof |
| contents | For an analytic function $ϕ(z)$ with a Laurent expansion at $\infty$ of the form
\begin{equation*}
ϕ(z)=z+c_{0}+\frac{c_{1}}{z}+\frac{c_{2}}{z^{2}}+\cdots,
\end{equation*} the Faber polynomial $F_n$ of degree $n$ associated to $ϕ$ is the polynomial part of the Laurent series at $\infty$ of $ϕ(z)^n$. We prove that the $n$th Chebyshev polynomial $T_{n,L_r}$ for the equipotential curve $L_r=\{z\in \mathbb{C}:|ϕ(z)|=r \}$ converges to $F_n$ as $r\to\infty$. The proof makes use of the fact that zero is the strongly unique best approximation to the monomial $z^n$ on the unit circle by polynomials of degree less than $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03967 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chebyshev polynomials on equipotential curves Miña-Díaz, Erwin Rubin, Olof Complex Variables 30C10, 30C20, 41A50, 30E10, 31A15 For an analytic function $ϕ(z)$ with a Laurent expansion at $\infty$ of the form \begin{equation*} ϕ(z)=z+c_{0}+\frac{c_{1}}{z}+\frac{c_{2}}{z^{2}}+\cdots, \end{equation*} the Faber polynomial $F_n$ of degree $n$ associated to $ϕ$ is the polynomial part of the Laurent series at $\infty$ of $ϕ(z)^n$. We prove that the $n$th Chebyshev polynomial $T_{n,L_r}$ for the equipotential curve $L_r=\{z\in \mathbb{C}:|ϕ(z)|=r \}$ converges to $F_n$ as $r\to\infty$. The proof makes use of the fact that zero is the strongly unique best approximation to the monomial $z^n$ on the unit circle by polynomials of degree less than $n$. |
| title | Chebyshev polynomials on equipotential curves |
| topic | Complex Variables 30C10, 30C20, 41A50, 30E10, 31A15 |
| url | https://arxiv.org/abs/2505.03967 |