Approximation of plurisubharmonic functions by logarithms of Gaussian analytic functions
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917962413244416 |
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| author | Eum, Kiyoon |
| author_facet | Eum, Kiyoon |
| contents | Let $Ω$ be a bounded pseudoconvex domain in $\mathbb{C}^N$. Given a continuous plurisubharmonic function $u$ on $Ω$, we construct a sequence of Gaussian analytic functions $f_n$ on $Ω$ associated with $u$ such that $\frac{1}{n}\log|f_n|$ converges to $u$ in $L^1_{loc}(Ω)$ almost surely, as $n\rightarrow\infty$. Gaussian analytic function $f_n$ is defined through its covariance, or equivalently, via its reproducing kernel Hilbert space, which corresponds to the weighted Bergman space with weight $e^{-2nu}$ with respect to the Lebesgue measure. As a consequence, we show the normalized zeros of $f_n$ converge to $dd^c u$ in the sense of currents. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11833 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximation of plurisubharmonic functions by logarithms of Gaussian analytic functions Eum, Kiyoon Complex Variables Probability 32U05, 60G15 Let $Ω$ be a bounded pseudoconvex domain in $\mathbb{C}^N$. Given a continuous plurisubharmonic function $u$ on $Ω$, we construct a sequence of Gaussian analytic functions $f_n$ on $Ω$ associated with $u$ such that $\frac{1}{n}\log|f_n|$ converges to $u$ in $L^1_{loc}(Ω)$ almost surely, as $n\rightarrow\infty$. Gaussian analytic function $f_n$ is defined through its covariance, or equivalently, via its reproducing kernel Hilbert space, which corresponds to the weighted Bergman space with weight $e^{-2nu}$ with respect to the Lebesgue measure. As a consequence, we show the normalized zeros of $f_n$ converge to $dd^c u$ in the sense of currents. |
| title | Approximation of plurisubharmonic functions by logarithms of Gaussian analytic functions |
| topic | Complex Variables Probability 32U05, 60G15 |
| url | https://arxiv.org/abs/2402.11833 |