Holomorphic approximation by polynomials with exponents restricted to a convex cone
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912515373400064 |
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| author | Sigurðardóttir, Álfheiður Edda |
| author_facet | Sigurðardóttir, Álfheiður Edda |
| contents | We study the approximation of holomorphic functions of several complex variables by the ring $\mathcal{P}^S(\mathbb{C}^n)$ of polynomials whose exponents are restricted to a convex cone $\mathbb{R}_+S$ for some compact convex $S\in \mathbb{R}^n_+$. We show a version of the Runge-Oka-Weil Theorem on approximation by these subrings on compact subsets of $\mathbb{C}^{*n}$ that are convex with respect to $\mathcal{P}^S(\mathbb{C}^n)$. We show a sharper result on rotationally symmetric compact sets. The tools used are Hörmander's $L^2$-theory and Siciak-Zakharyuta functions $V^S_K$ associated to $S$. We provide a formula for $V^S_K$ when $K$ is a rotationally symmetric compact subset of $\mathbb{C}^{*n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12132 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Holomorphic approximation by polynomials with exponents restricted to a convex cone Sigurðardóttir, Álfheiður Edda Complex Variables 32A08 (Primary) 32A10 (Secondary) We study the approximation of holomorphic functions of several complex variables by the ring $\mathcal{P}^S(\mathbb{C}^n)$ of polynomials whose exponents are restricted to a convex cone $\mathbb{R}_+S$ for some compact convex $S\in \mathbb{R}^n_+$. We show a version of the Runge-Oka-Weil Theorem on approximation by these subrings on compact subsets of $\mathbb{C}^{*n}$ that are convex with respect to $\mathcal{P}^S(\mathbb{C}^n)$. We show a sharper result on rotationally symmetric compact sets. The tools used are Hörmander's $L^2$-theory and Siciak-Zakharyuta functions $V^S_K$ associated to $S$. We provide a formula for $V^S_K$ when $K$ is a rotationally symmetric compact subset of $\mathbb{C}^{*n}$. |
| title | Holomorphic approximation by polynomials with exponents restricted to a convex cone |
| topic | Complex Variables 32A08 (Primary) 32A10 (Secondary) |
| url | https://arxiv.org/abs/2409.12132 |