Quantitative Constraints for Stable Sampling on the Sphere
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909983786926080 |
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| author | Ehler, Martin Gröchenig, Karlheinz |
| author_facet | Ehler, Martin Gröchenig, Karlheinz |
| contents | We derive quantitative volume constraints for sampling measures $μ_t$ on the unit sphere $\mathbb{S}^d$ that satisfy Marcinkiewicz-Zygmund inequalities of order $t$. Using precise localization estimates for Jacobi polynomials, we obtain explicit upper and lower bounds on the $μ_t$-mass of geodesic balls at the natural scale $t^{-1}$. Whereas constants are typically left implicit in the literature, we place special emphasis on fully explicit constants, and the results are genuinely quantitative. Moreover, these bounds yield quantitative constraints for the $s$-dimensional Hausdorff volume of Marcinkiewicz-Zygmund sampling sets and, in particular, optimal lower bounds for the length of Marcinkiewicz-Zygmund curves. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_04119 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantitative Constraints for Stable Sampling on the Sphere Ehler, Martin Gröchenig, Karlheinz Numerical Analysis 41A55, 41A63, 94A12, 26B15 We derive quantitative volume constraints for sampling measures $μ_t$ on the unit sphere $\mathbb{S}^d$ that satisfy Marcinkiewicz-Zygmund inequalities of order $t$. Using precise localization estimates for Jacobi polynomials, we obtain explicit upper and lower bounds on the $μ_t$-mass of geodesic balls at the natural scale $t^{-1}$. Whereas constants are typically left implicit in the literature, we place special emphasis on fully explicit constants, and the results are genuinely quantitative. Moreover, these bounds yield quantitative constraints for the $s$-dimensional Hausdorff volume of Marcinkiewicz-Zygmund sampling sets and, in particular, optimal lower bounds for the length of Marcinkiewicz-Zygmund curves. |
| title | Quantitative Constraints for Stable Sampling on the Sphere |
| topic | Numerical Analysis 41A55, 41A63, 94A12, 26B15 |
| url | https://arxiv.org/abs/2601.04119 |