Scaled global operators and Fueter variables on non-zero scaled hypercomplex numbers
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| Format: | Preprint |
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2024
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| _version_ | 1866914740703330304 |
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| author | Alpay, Daniel Cho, Ilwoo Vajiac, Mihaela |
| author_facet | Alpay, Daniel Cho, Ilwoo Vajiac, Mihaela |
| contents | In this paper we describe the rise of global operators in the scaled quaternionic case, an important extension from the quaternionic case to the family of scaled hypercomplex numbers $\mathbb{H}_t,\, t\in\mathbb{R}^*$, of which the $\mathbb{H}_{-1}=\mathbb{H}$ is the space of quaternions and $\mathbb{H}_{1}$ is the space of split quaternions. We also describe the scaled Fueter-type variables associated to these operators, developing a coherent theory in this field. We use these types of variables to build different types of function spaces on $\mathbb{H}_t$. Counterparts of the Hardy space and of the Arveson space are also introduced and studied in the present setting. The two different adjoints in the scaled hypercomplex numbers lead to two parallel cases in each instance. Finally we introduce and study the notion of rational function. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_03065 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Scaled global operators and Fueter variables on non-zero scaled hypercomplex numbers Alpay, Daniel Cho, Ilwoo Vajiac, Mihaela Functional Analysis Mathematical Physics 30G35, 46E22 In this paper we describe the rise of global operators in the scaled quaternionic case, an important extension from the quaternionic case to the family of scaled hypercomplex numbers $\mathbb{H}_t,\, t\in\mathbb{R}^*$, of which the $\mathbb{H}_{-1}=\mathbb{H}$ is the space of quaternions and $\mathbb{H}_{1}$ is the space of split quaternions. We also describe the scaled Fueter-type variables associated to these operators, developing a coherent theory in this field. We use these types of variables to build different types of function spaces on $\mathbb{H}_t$. Counterparts of the Hardy space and of the Arveson space are also introduced and studied in the present setting. The two different adjoints in the scaled hypercomplex numbers lead to two parallel cases in each instance. Finally we introduce and study the notion of rational function. |
| title | Scaled global operators and Fueter variables on non-zero scaled hypercomplex numbers |
| topic | Functional Analysis Mathematical Physics 30G35, 46E22 |
| url | https://arxiv.org/abs/2404.03065 |