Quantitative Constraints for Stable Sampling on the Sphere
Martin Ehler, Karlheinz Gröchenig
TL;DR
This work provides fully explicit, dimension-dependent quantitative constraints for sampling on the unit sphere $\mathbb{S}^d$ via Marcinkiewicz-Zygmund inequalities of order $t$. By exploiting quantitative localization of reproducing kernels generated from Jacobi polynomials at scale $t^{-1}$, the authors derive explicit upper and lower bounds on the $\mu_t$-mass of geodesic balls and deduce volume constraints for $s$-dimensional sampling sets and lower bounds on the length of Marcinkiewicz-Zygmund curves. The results extend beyond point sets to curves and higher-dimensional subsets, yielding lower bounds on Hausdorff measures with explicit constants and establishing off-diagonal kernel decay to support lower-volume bounds. The contributions provide practical, scalable criteria for stable spherical sampling with provable constants, relevant to numerical quadrature, spherical $t$-designs, and curve-based data acquisition. Overall, the paper delivers a genuinely quantitative, highly explicit framework connecting kernel localization, sampling inequalities, and geometric constraints on sampling on spheres of arbitrary dimension.
Abstract
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.