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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2410.20985 |
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| _version_ | 1866913815462936576 |
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| author | Calzi, Mattia |
| author_facet | Calzi, Mattia |
| contents | Given a bounded symmetric domain $D$ in $\mathbb C^n$, we consider the Clark measures $μ_α$, $α\in \mathbb T$, associated with a rational inner function $φ$ from $D$ into the unit disc in $\mathbb C$. We show that $μ_α=c|\nabla φ|^{-1}χ_{\mathrm b D \cap φ^{-1}(α)}\cdot \mathcal H^{m-1}$, where $m$ is the dimension of the Shilov boundary $\mathrm b D$ of $D$ and $c$ is a suitable constant. Denoting with $H^2(μ_α)$ the closure of the space of holomorphic polynomials in $L^2(μ_α)$, we characterize the $α$ for which $H^2(μ_α)=L^2(μ_α)$ when $D$ is a polydisc; we also provide some necessary and some sufficient conditions for general domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20985 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Clark Measures Associated with Rational Inner Functions on Bounded Symmetric Domains Calzi, Mattia Complex Variables 32M15, 32A08 Given a bounded symmetric domain $D$ in $\mathbb C^n$, we consider the Clark measures $μ_α$, $α\in \mathbb T$, associated with a rational inner function $φ$ from $D$ into the unit disc in $\mathbb C$. We show that $μ_α=c|\nabla φ|^{-1}χ_{\mathrm b D \cap φ^{-1}(α)}\cdot \mathcal H^{m-1}$, where $m$ is the dimension of the Shilov boundary $\mathrm b D$ of $D$ and $c$ is a suitable constant. Denoting with $H^2(μ_α)$ the closure of the space of holomorphic polynomials in $L^2(μ_α)$, we characterize the $α$ for which $H^2(μ_α)=L^2(μ_α)$ when $D$ is a polydisc; we also provide some necessary and some sufficient conditions for general domains. |
| title | Clark Measures Associated with Rational Inner Functions on Bounded Symmetric Domains |
| topic | Complex Variables 32M15, 32A08 |
| url | https://arxiv.org/abs/2410.20985 |