Reduced-quaternion inframonogenic functions on the ball
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866913534690983936 |
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| author | Álvarez, C. Morais, J. Porter, R. Michael |
| author_facet | Álvarez, C. Morais, J. Porter, R. Michael |
| contents | A function $f$ from a domain in $\mathbb{R}^3$ to the quaternions is said to be inframonogenic if $\overline{\partial}\, f\overline{\partial} =0$, where $\overline{\partial} = \partial/\partial x_0+ (\partial/\partial x_1)e_1+(\partial/\partial x_2) e_2$. All inframonogenic functions are biharmonic. In the context of functions $f=f_0+f_1e_1+f_2e_2$ taking values in the reduced quaternions, we show that the homogeneous polynomials of degree $n$ form a subspace of dimension $6n+3$. We use them to construct an explicit, computable orthogonal basis for the Hilbert space of square-integrable inframonogenic functions defined in the ball in $\mathbb{R}^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_04509 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Reduced-quaternion inframonogenic functions on the ball Álvarez, C. Morais, J. Porter, R. Michael Complex Variables Primary 30G35, Secondary 31B30, 33D45, 35G05, 42C30 A function $f$ from a domain in $\mathbb{R}^3$ to the quaternions is said to be inframonogenic if $\overline{\partial}\, f\overline{\partial} =0$, where $\overline{\partial} = \partial/\partial x_0+ (\partial/\partial x_1)e_1+(\partial/\partial x_2) e_2$. All inframonogenic functions are biharmonic. In the context of functions $f=f_0+f_1e_1+f_2e_2$ taking values in the reduced quaternions, we show that the homogeneous polynomials of degree $n$ form a subspace of dimension $6n+3$. We use them to construct an explicit, computable orthogonal basis for the Hilbert space of square-integrable inframonogenic functions defined in the ball in $\mathbb{R}^3$. |
| title | Reduced-quaternion inframonogenic functions on the ball |
| topic | Complex Variables Primary 30G35, Secondary 31B30, 33D45, 35G05, 42C30 |
| url | https://arxiv.org/abs/2203.04509 |