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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.00403 |
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| _version_ | 1866910206953259008 |
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| author | Ariwahjoedi, Seramika Rosyid, Muhammad Farchani Wijaya, Andika Kusuma |
| author_facet | Ariwahjoedi, Seramika Rosyid, Muhammad Farchani Wijaya, Andika Kusuma |
| contents | We extend Fourier analysis to curved spaces by defining a Generalized Fourier Transform (GFT) on any Riemannian manifold $Σ$ via spectral decomposition. Under minimal requirements that the transform is an isometric isomorphism and has a kernel diagonalizing the Laplace-Beltrami operator, we prove that the GFT satisfies a generalized Parseval-Plancherel theorem. To resolve the spectral degeneracy that obscures "momentum space" in such settings, we require the degenerate sector to be resolved by a local, symmetry-adapted maximal Abelian commuting set (a fiberwise MASA), constructed from geometric differential operators, most notably from Killing data when such symmetries are available. We provide a constructive algorithm for generating these commuting operators and show that the resulting momentum label spaces $\mathcal{F}$ (discrete, continuous, or mixed) reflect geometric symmetry constraints. We introduce a dual classification: (i) by MASA completeness and Stackel separability, and (ii) by the topology of $\mathcal{F}$. Finally, we distinguish unitary changes induced by true isometries (which preserve the GFT structure) from changes of coordinate-adapted degeneracy resolution/separation schemes, which may induce inequivalent $k$-space labelings (e.g. Cartesian vs spherical constructions in $\mathbb{R}^{3}$) while remaining unitarily equivalent on $\mathcal{L}^{2}\left[Σ\right]$. This symmetry-adapted harmonic analysis is intended as a foundation for curved-space mode decompositions; dynamical applications are developed in the subsequent work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_00403 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generalized Fourier Transforms for Momentum-Space Construction on Riemannian Manifolds Ariwahjoedi, Seramika Rosyid, Muhammad Farchani Wijaya, Andika Kusuma Mathematical Physics We extend Fourier analysis to curved spaces by defining a Generalized Fourier Transform (GFT) on any Riemannian manifold $Σ$ via spectral decomposition. Under minimal requirements that the transform is an isometric isomorphism and has a kernel diagonalizing the Laplace-Beltrami operator, we prove that the GFT satisfies a generalized Parseval-Plancherel theorem. To resolve the spectral degeneracy that obscures "momentum space" in such settings, we require the degenerate sector to be resolved by a local, symmetry-adapted maximal Abelian commuting set (a fiberwise MASA), constructed from geometric differential operators, most notably from Killing data when such symmetries are available. We provide a constructive algorithm for generating these commuting operators and show that the resulting momentum label spaces $\mathcal{F}$ (discrete, continuous, or mixed) reflect geometric symmetry constraints. We introduce a dual classification: (i) by MASA completeness and Stackel separability, and (ii) by the topology of $\mathcal{F}$. Finally, we distinguish unitary changes induced by true isometries (which preserve the GFT structure) from changes of coordinate-adapted degeneracy resolution/separation schemes, which may induce inequivalent $k$-space labelings (e.g. Cartesian vs spherical constructions in $\mathbb{R}^{3}$) while remaining unitarily equivalent on $\mathcal{L}^{2}\left[Σ\right]$. This symmetry-adapted harmonic analysis is intended as a foundation for curved-space mode decompositions; dynamical applications are developed in the subsequent work. |
| title | Generalized Fourier Transforms for Momentum-Space Construction on Riemannian Manifolds |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2605.00403 |