Optimal asymptotic bounds for designs on manifolds
arXiv:1811.1267623 citationsh-index: 14
Originality Incremental advance
AI Analysis
Provides optimal asymptotic bounds for designs on general manifolds, solving a fundamental problem in approximation theory and numerical integration.
The paper proves that for any compact connected oriented Riemannian manifold, there exist L-designs with N nodes for any N ≥ C_M L^d, extending a known result from the sphere to general manifolds.
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 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.