Directional polynomial frames on spheres
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914272860176384 |
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| author | Hasannasab, Marzieh Kaldewey, Larissa Schoppert, Frederic |
| author_facet | Hasannasab, Marzieh Kaldewey, Larissa Schoppert, Frederic |
| contents | We introduce a general framework for the construction of polynomial frames in $L^2(\mathbb{S}^{d-1})$, $d \geq 3$, where the frame functions are obtained as rotated versions of an initial sequence of polynomials $Ψ^j$, $j\in \mathbb{N}_0$. The rotations involved are discretized using suitable quadrature rules. This framework includes classical constructions such as spherical needlets and directional wavelet systems, and at the same time permits the systematic design of new frames with adjustable spatial localization, directional sensitivity, and computational complexity. We show that a number of frame properties can be characterized in terms of simple, easily verifiable conditions on the Fourier coefficients of the functions $Ψ^j$. Extending an earlier result for zonal systems, we establish sufficient conditions under which the frame functions are optimally localized in space with respect to a spherical uncertainty principle, thus making the corresponding systems a viable tool for position-frequency analyses. To conclude this article, we explicitly discuss examples of well-localized and highly directional polynomial frames. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_15883 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Directional polynomial frames on spheres Hasannasab, Marzieh Kaldewey, Larissa Schoppert, Frederic Classical Analysis and ODEs Functional Analysis 42C15, 42C40, 65T60 We introduce a general framework for the construction of polynomial frames in $L^2(\mathbb{S}^{d-1})$, $d \geq 3$, where the frame functions are obtained as rotated versions of an initial sequence of polynomials $Ψ^j$, $j\in \mathbb{N}_0$. The rotations involved are discretized using suitable quadrature rules. This framework includes classical constructions such as spherical needlets and directional wavelet systems, and at the same time permits the systematic design of new frames with adjustable spatial localization, directional sensitivity, and computational complexity. We show that a number of frame properties can be characterized in terms of simple, easily verifiable conditions on the Fourier coefficients of the functions $Ψ^j$. Extending an earlier result for zonal systems, we establish sufficient conditions under which the frame functions are optimally localized in space with respect to a spherical uncertainty principle, thus making the corresponding systems a viable tool for position-frequency analyses. To conclude this article, we explicitly discuss examples of well-localized and highly directional polynomial frames. |
| title | Directional polynomial frames on spheres |
| topic | Classical Analysis and ODEs Functional Analysis 42C15, 42C40, 65T60 |
| url | https://arxiv.org/abs/2601.15883 |