A representation formula for slice regular functions over slice-cones in several variables
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2020
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| _version_ | 1866910287209168896 |
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| author | Dou, Xinyuan Ren, Guangbin Sabadini, Irene |
| author_facet | Dou, Xinyuan Ren, Guangbin Sabadini, Irene |
| contents | The aim of this paper is to extend the so called slice analysis to a general case in which the codomain is a real vector space of even dimension, i.e. is of the form $\mathbb{R}^{2n}$. We define a cone $\mathcal{W}_\mathcal{C}^d$ in $[End(\mathbb{R}^{2n})]^d$ and we extend the slice-topology $τ_s$ to this cone. Slice regular functions can be defined on open sets in $\left(τ_s,\mathcal{W}_\mathcal{C}^d\right)$ and a number of results can be proved in this framework, among which a representation formula. This theory can be applied to some real algebras, called left slice complex structure algebras. These algebras include quaternions, octonions, Clifford algebras and real alternative $*$-algebras but also left-alternative algebras and sedenions, thus providing brand new settings in slice analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_13770 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A representation formula for slice regular functions over slice-cones in several variables Dou, Xinyuan Ren, Guangbin Sabadini, Irene Complex Variables 30G35, 17A30 The aim of this paper is to extend the so called slice analysis to a general case in which the codomain is a real vector space of even dimension, i.e. is of the form $\mathbb{R}^{2n}$. We define a cone $\mathcal{W}_\mathcal{C}^d$ in $[End(\mathbb{R}^{2n})]^d$ and we extend the slice-topology $τ_s$ to this cone. Slice regular functions can be defined on open sets in $\left(τ_s,\mathcal{W}_\mathcal{C}^d\right)$ and a number of results can be proved in this framework, among which a representation formula. This theory can be applied to some real algebras, called left slice complex structure algebras. These algebras include quaternions, octonions, Clifford algebras and real alternative $*$-algebras but also left-alternative algebras and sedenions, thus providing brand new settings in slice analysis. |
| title | A representation formula for slice regular functions over slice-cones in several variables |
| topic | Complex Variables 30G35, 17A30 |
| url | https://arxiv.org/abs/2011.13770 |