Polynomials with exponents in compact convex sets and associated weighted extremal functions -- The Bernstein-Walsh-Siciak theorem
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912091794833408 |
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| author | Magnússon, Benedikt Steinar Sigurðsson, Ragnar Snorrason, Bergur |
| author_facet | Magnússon, Benedikt Steinar Sigurðsson, Ragnar Snorrason, Bergur |
| contents | We generalize the Bernstein-Walsh-Siciak theorem on polynomial approximation in $\mathbb{C}^n$ to the case where the polynomial ring $\mathcal{P}(\mathbb{C}^n)$ is replaced by a subring $\mathcal{P}^S(\mathbb{C}^n)$ consisting of all polynomials with exponents restricted to sets $mS$, where $S$ is a compact convex subset of $\mathbb{R}_+^n$ with $0 \in S$ and $m = 0, 1, 2, 3, \dots$, and uniform estimates of error in the approximation are replaced by weighted uniform estimates with respect to an admissible weight function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_02486 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Polynomials with exponents in compact convex sets and associated weighted extremal functions -- The Bernstein-Walsh-Siciak theorem Magnússon, Benedikt Steinar Sigurðsson, Ragnar Snorrason, Bergur Complex Variables 32U35 (Primary) 32A08, 32A15, 32U15, 32W05 (Secondary) We generalize the Bernstein-Walsh-Siciak theorem on polynomial approximation in $\mathbb{C}^n$ to the case where the polynomial ring $\mathcal{P}(\mathbb{C}^n)$ is replaced by a subring $\mathcal{P}^S(\mathbb{C}^n)$ consisting of all polynomials with exponents restricted to sets $mS$, where $S$ is a compact convex subset of $\mathbb{R}_+^n$ with $0 \in S$ and $m = 0, 1, 2, 3, \dots$, and uniform estimates of error in the approximation are replaced by weighted uniform estimates with respect to an admissible weight function. |
| title | Polynomials with exponents in compact convex sets and associated weighted extremal functions -- The Bernstein-Walsh-Siciak theorem |
| topic | Complex Variables 32U35 (Primary) 32A08, 32A15, 32U15, 32W05 (Secondary) |
| url | https://arxiv.org/abs/2306.02486 |