#### Vol. 14, No. 6, 2021

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Optimal asymptotic bounds for designs on manifolds

### Bianca Gariboldi and Giacomo Gigante

Vol. 14 (2021), No. 6, 1701–1724
##### Abstract

We extend to the case of a $d$-dimensional compact connected oriented Riemannian manifold $\mathsc{ℳ}$ 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 $N\ge {C}_{\mathsc{ℳ}}{L}^{d}$. For this, we need to prove a version of the Marcinkiewicz–Zygmund inequality for the gradient of diffusion polynomials.

##### Keywords
designs, Riemannian manifolds, Marcinkiewicz–Zygmund inequalities
##### Mathematical Subject Classification 2010
Primary: 41A55, 42C15
Secondary: 58J35
##### Milestones
Received: 21 December 2018
Revised: 14 January 2020
Accepted: 19 March 2020
Published: 7 September 2021
##### Authors
 Bianca Gariboldi Dipartimento di Ingegneria Gestionale, dell’Informazione e della Produzione Università degli Studi di Bergamo Dalmine Italy Giacomo Gigante Dipartimento di Ingegneria Gestionale, dell’Informazione e della Produzione Università degli Studi di Bergamo Dalmine Italy