Szegő's theorem for Jordan arcs
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909790101307392 |
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| author | Buchecker, Benedikt |
| author_facet | Buchecker, Benedikt |
| contents | The $n$-th Christoffel function for a point $z_0\in\mathbb C$ and a finite measure $μ$ supported on a Jordan arc $Γ$ is \[ λ_n(μ,z_0)=\inf\left\{\int_Γ|P|^2dμ\mid P\text{ is a polynomial of degree at most }n\text{ and } P(z_0)=1\right\}. \] It is natural to extend this notion to $z_0=\infty$ and define $λ_n(μ,\infty)$ to be the infimum of the squared $L^2(μ)$-norm over monic polynomials of degree $n$. The classical Szegő theorem provides an asymptotic description of $λ_n(μ,z_0)$ for $|z_0|>1$ and $z_0=\infty$ and arbitrary finite measures supported on the unit circle. Widom has proved a version of Szegő's theorem for measures supported on $C^{2+}$-Jordan arcs for the point $z_0=\infty$ and purely absolutely continuous measures belonging to the Szegő class. We extend this result in two directions. We prove explicit asymptotics of $λ_n(μ,z_0)$ for any finite measure $μ$ supported on a $C^{1+}$-Jordan arc $Γ$, and for all points $z_0\in\mathbb{C}\cup\{\infty\}\setminusΓ$. Moreover, if the measure is in the Szegő class, we provide explicit asymptotics for the extremal and orthogonal polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12445 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Szegő's theorem for Jordan arcs Buchecker, Benedikt Classical Analysis and ODEs 30C10, 30E15, 42C05, 30C85 The $n$-th Christoffel function for a point $z_0\in\mathbb C$ and a finite measure $μ$ supported on a Jordan arc $Γ$ is \[ λ_n(μ,z_0)=\inf\left\{\int_Γ|P|^2dμ\mid P\text{ is a polynomial of degree at most }n\text{ and } P(z_0)=1\right\}. \] It is natural to extend this notion to $z_0=\infty$ and define $λ_n(μ,\infty)$ to be the infimum of the squared $L^2(μ)$-norm over monic polynomials of degree $n$. The classical Szegő theorem provides an asymptotic description of $λ_n(μ,z_0)$ for $|z_0|>1$ and $z_0=\infty$ and arbitrary finite measures supported on the unit circle. Widom has proved a version of Szegő's theorem for measures supported on $C^{2+}$-Jordan arcs for the point $z_0=\infty$ and purely absolutely continuous measures belonging to the Szegő class. We extend this result in two directions. We prove explicit asymptotics of $λ_n(μ,z_0)$ for any finite measure $μ$ supported on a $C^{1+}$-Jordan arc $Γ$, and for all points $z_0\in\mathbb{C}\cup\{\infty\}\setminusΓ$. Moreover, if the measure is in the Szegő class, we provide explicit asymptotics for the extremal and orthogonal polynomials. |
| title | Szegő's theorem for Jordan arcs |
| topic | Classical Analysis and ODEs 30C10, 30E15, 42C05, 30C85 |
| url | https://arxiv.org/abs/2509.12445 |