Quantitative Constraints for Stable Sampling on the Sphere

TL;DR

This paper establishes explicit quantitative volume constraints for sampling measures on the sphere that satisfy Marcinkiewicz-Zygmund inequalities, providing precise bounds and implications for sampling set sizes.

Contribution

It introduces fully explicit constants and bounds for sampling measures on the sphere satisfying Marcinkiewicz-Zygmund inequalities, enhancing the quantitative understanding of sampling distributions.

Findings

Explicit upper and lower bounds on measure of geodesic balls at scale t^{-1}

Quantitative constraints for Hausdorff volume of sampling sets

Optimal lower bounds for length of Marcinkiewicz-Zygmund curves

Abstract

We derive quantitative volume constraints for sampling measures $\mu_t$ 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 $\mu_t$-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.