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Comptes Rendus Mathématique
Volume 355, n° 7
pages 780-785 (juillet 2017)
Doi : 10.1016/j.crma.2017.06.004
Received : 1 June 2017 ;  accepted : 7 June 2017
Rigidity of optimal bases for signal spaces
Rigidité des bases optimales pour les espaces de signaux
 

Haïm Brezis a, b, c , David Gómez-Castro d, e
a Department of Mathematics, Hill Center, Busch Campus, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA 
b Departments of Mathematics and Computer Science, Technion, Israel Institute of Technology, 32000 Haifa, Israel 
c Laboratoire Jacques-Louis-Lions, Université Pierre-et-Marie-Curie, 4, place Jussieu, 75252 Paris cedex 05, France 
d Dpto. de Matemática Aplicada, Universidad Complutense de Madrid, Spain 
e Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, Spain 

Abstract

We discuss optimal  -approximations of functions controlled in the  -norm. We prove that the basis of eigenfunctions of the Laplace operator with Dirichlet boundary condition is the only orthonormal basis   of   that provides an optimal approximation in the sense of
‖f−∑i=1n(f,bi)bi‖L22≤‖∇f‖L22λn+1∀f∈H01(Ω),∀n≥1. This solves an open problem raised by Y. Aflalo, H. Brezis, A. Bruckstein, R. Kimmel, and N. Sochen (Best bases for signal spaces, C. R. Acad. Sci. Paris, Ser. I 354 (12) (2016) 1155–1167).

The full text of this article is available in PDF format.
Résumé

On s'intéresse à l'approximation optimale pour la norme   de fonctions contrôlées en norme  . On prouve que la base des fonctions propres du laplacien avec condition de Dirichlet au bord est l'unique base orthonormale   de   qui réalise une approximation optimale au sens de
‖f−∑i=1n(f,bi)bi‖L22≤‖∇f‖L22λn+1∀f∈H01(Ω),∀n≥1. Ceci résout un problème ouvert posé par Y. Aflalo, H. Brezis, A. Bruckstein, R. Kimmel et N. Sochen (Best bases for signal spaces, C. R. Acad. Sci. Paris, Ser. I 354 (12) (2016) 1155–1167).

The full text of this article is available in PDF format.


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