Optimal asymptotic bounds for designs on manifolds

Gariboldi, Bianca; Gigante, Giacomo

doi:10.2140/apde.2021.14.1701

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Optimal asymptotic bounds for designs on manifolds

Bianca Gariboldi and Giacomo Gigante

Vol. 14 (2021), No. 6, 1701–1724

DOI: 10.2140/apde.2021.14.1701

Abstract

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

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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

Vol. 14, No. 6, 2021