Asymptotically short generalizations of $t$-design curves
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918010975944704 |
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| author | Lindblad, Ayodeji |
| author_facet | Lindblad, Ayodeji |
| contents | Ehler and Gröchenig posed the question of finding $t$-design curves $γ_t$$\unicode{x2013}$curves whose associated line integrals exactly average all degree at most $t$ polynomials$\unicode{x2013}$on $S^d$ of asymptotically optimal arc length $\ell(γ_t)\asymp t^{d-1}$ as $t\to\infty$. This work investigates analogues of this question for $\textit{weighted}$ and $\textit{$\varepsilon_t$-approximate $t$-design curves}$, proving existence of such curves $γ_t$ on $S^d$ of arc length $\ell(γ_t)\asymp t^{d-1}$ as $t\to\infty$ for all $d\in\Bbb N_+$ in the weighted setting (in which case such curves are asymptotically optimal) and all odd $d\in\Bbb N_+$ in the approximate setting (where we have $\varepsilon_t\asymp1/t$ as $t\to\infty$). Formulas for such weighted $t$-design curves for $d\in\{2,3\}$ are presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03056 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotically short generalizations of $t$-design curves Lindblad, Ayodeji Metric Geometry Numerical Analysis 41A55 (primary), 05B30, 41A63, 41A81, 65D30, 65D32 (secondary) Ehler and Gröchenig posed the question of finding $t$-design curves $γ_t$$\unicode{x2013}$curves whose associated line integrals exactly average all degree at most $t$ polynomials$\unicode{x2013}$on $S^d$ of asymptotically optimal arc length $\ell(γ_t)\asymp t^{d-1}$ as $t\to\infty$. This work investigates analogues of this question for $\textit{weighted}$ and $\textit{$\varepsilon_t$-approximate $t$-design curves}$, proving existence of such curves $γ_t$ on $S^d$ of arc length $\ell(γ_t)\asymp t^{d-1}$ as $t\to\infty$ for all $d\in\Bbb N_+$ in the weighted setting (in which case such curves are asymptotically optimal) and all odd $d\in\Bbb N_+$ in the approximate setting (where we have $\varepsilon_t\asymp1/t$ as $t\to\infty$). Formulas for such weighted $t$-design curves for $d\in\{2,3\}$ are presented. |
| title | Asymptotically short generalizations of $t$-design curves |
| topic | Metric Geometry Numerical Analysis 41A55 (primary), 05B30, 41A63, 41A81, 65D30, 65D32 (secondary) |
| url | https://arxiv.org/abs/2505.03056 |