Optimal asymptotic bounds for designs on manifolds

Abstract

We extend to the case of a $d$-dimensional compact connected oriented Riemannian manifold $\mathcal{ℳ}$ the theorem of A. Bondarenko, D. Radchenko and M. Viazovska (Ann. of Math. $\left(2\right)$ 178:2 (2013), 443–452) on the existence of $L$-designs consisting of $N$ nodes for any $N\ge C_{\mathcal{ℳ}}{L}^d$. For this, we need to prove a version of the Marcinkiewicz–Zygmund inequality for the gradient of diffusion polynomials.