Non-Abelian Fourier Analysis on $\mathbfΓ\backslash SE(d)$
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911810797436928 |
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| author | Farashahi, Arash Ghaani Chirikjian, Gregory S. |
| author_facet | Farashahi, Arash Ghaani Chirikjian, Gregory S. |
| contents | This paper presents a systematic study for the general theory of non-Abelian Fourier series of integrable functions on the homogeneous space $\mathbfΓ\backslash SE(d)$, where $SE(d)$ is the special Euclidean group in dimension $d$, and $\mathbf{\mathbfΓ}$ is a discrete and co-compact subgroup of $SE(d)$. Suppose that $μ$ is the finite $SE(d)$-invariant measure on the right coset space $\mathbfΓ\backslash SE(d)$, normalized with respect to Weil's formula. The analytic aspects of the proposed method works for any given orthonormal basis of the Hilbert function space $L^2(\mathbfΓ\backslash SE(d),μ)$. The paper is concluded with some convolution and Plancherel formulas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15874 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-Abelian Fourier Analysis on $\mathbfΓ\backslash SE(d)$ Farashahi, Arash Ghaani Chirikjian, Gregory S. Functional Analysis Primary 43A30, 43A85, Secondary 20H15, 43A10, 43A15, 43A20, 68T40, 74E15, 82D25 This paper presents a systematic study for the general theory of non-Abelian Fourier series of integrable functions on the homogeneous space $\mathbfΓ\backslash SE(d)$, where $SE(d)$ is the special Euclidean group in dimension $d$, and $\mathbf{\mathbfΓ}$ is a discrete and co-compact subgroup of $SE(d)$. Suppose that $μ$ is the finite $SE(d)$-invariant measure on the right coset space $\mathbfΓ\backslash SE(d)$, normalized with respect to Weil's formula. The analytic aspects of the proposed method works for any given orthonormal basis of the Hilbert function space $L^2(\mathbfΓ\backslash SE(d),μ)$. The paper is concluded with some convolution and Plancherel formulas. |
| title | Non-Abelian Fourier Analysis on $\mathbfΓ\backslash SE(d)$ |
| topic | Functional Analysis Primary 43A30, 43A85, Secondary 20H15, 43A10, 43A15, 43A20, 68T40, 74E15, 82D25 |
| url | https://arxiv.org/abs/2403.15874 |