We show how to build a kernel K_X(x, y) = \Sigma_(m=0)^X h(lambda_(m)/lambda_(X))phi_(m)(x)phi_(m)(y) on a compact Riemannian manifold M, which is positive up to a negligible error and such that K_X(x, x) approximate to X. Here 0 = lambda_(0) <= lambda_(1) <= ... are the eigenvalues of the Laplace-Beltrami operator on M, listed with repetitions, and phi_(0), phi_(1), ... an associated system of eigenfunctions, forming an orthonormal basis of L^2(M). The function h is smooth up to a certain minimal degree, even, compactly supported in [-1, 1] with h(0) = 1, and K_X(x, y) turns out to be an approximation to the identity.

(2022). Almost positive kernels on compact Riemannian manifolds [journal article - articolo]. In MATHEMATISCHE ZEITSCHRIFT. Retrieved from http://hdl.handle.net/10446/226849

Almost positive kernels on compact Riemannian manifolds

Gariboldi, Bianca;Gigante, Giacomo
2022

Abstract

We show how to build a kernel K_X(x, y) = \Sigma_(m=0)^X h(lambda_(m)/lambda_(X))phi_(m)(x)phi_(m)(y) on a compact Riemannian manifold M, which is positive up to a negligible error and such that K_X(x, x) approximate to X. Here 0 = lambda_(0) <= lambda_(1) <= ... are the eigenvalues of the Laplace-Beltrami operator on M, listed with repetitions, and phi_(0), phi_(1), ... an associated system of eigenfunctions, forming an orthonormal basis of L^2(M). The function h is smooth up to a certain minimal degree, even, compactly supported in [-1, 1] with h(0) = 1, and K_X(x, y) turns out to be an approximation to the identity.
articolo
Gariboldi, Bianca; Gigante, Giacomo
(2022). Almost positive kernels on compact Riemannian manifolds [journal article - articolo]. In MATHEMATISCHE ZEITSCHRIFT. Retrieved from http://hdl.handle.net/10446/226849
File allegato/i alla scheda:
File Dimensione del file Formato  
Gariboldi-Gigante2022_Article_AlmostPositiveKernelsOnCompact.pdf

accesso aperto

Versione: publisher's version - versione editoriale
Licenza: Creative commons
Dimensione del file 312.68 kB
Formato Adobe PDF
312.68 kB Adobe PDF Visualizza/Apri
Pubblicazioni consigliate

Caricamento pubblicazioni consigliate

Aisberg ©2008 Servizi bibliotecari, Università degli studi di Bergamo | Terms of use/Condizioni di utilizzo

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10446/226849
Citazioni
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 0
social impact