Rigidity of optimal bases for signal spaces

Haïm Brezis; David Gómez-Castro

doi:10.1016/j.crma.2017.06.004

Partial differential equations/Theory of signals

Rigidity of optimal bases for signal spaces

Presented by: Haïm Brezis

Haïm Brezis ^1,^2,³; David Gómez-Castro ^4,⁵

¹ Department of Mathematics, Hill Center, Busch Campus, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA
² Departments of Mathematics and Computer Science, Technion, Israel Institute of Technology, 32000 Haifa, Israel
³ Laboratoire Jacques-Louis-Lions, Université Pierre-et-Marie-Curie, 4, place Jussieu, 75252 Paris cedex 05, France
⁴ Dpto. de Matemática Aplicada, Universidad Complutense de Madrid, Spain
⁵ Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, Spain

Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 780-785.

Abstract
Résumé

We discuss optimal $L^{2}$ -approximations of functions controlled in the $H^{1}$ -norm. We prove that the basis of eigenfunctions of the Laplace operator with Dirichlet boundary condition is the only orthonormal basis $(b_{i})$ of $L^{2}$ that provides an optimal approximation in the sense of

{‖ f - \sum_{i = 1}^{n} (f, b_{i}) b_{i} ‖}_{L^{2}}^{2} \leq \frac{{‖ \nabla f ‖}_{L^{2}}^{2}}{λ_{n + 1}} \forall f \in H_{0}^{1} (Ω), \forall n \geq 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).

On s'intéresse à l'approximation optimale pour la norme $L^{2}$ de fonctions contrôlées en norme $H^{1}$ . On prouve que la base des fonctions propres du laplacien avec condition de Dirichlet au bord est l'unique base orthonormale $(b_{i})$ de $L^{2}$ qui réalise une approximation optimale au sens de

{‖ f - \sum_{i = 1}^{n} (f, b_{i}) b_{i} ‖}_{L^{2}}^{2} \leq \frac{{‖ \nabla f ‖}_{L^{2}}^{2}}{λ_{n + 1}} \forall f \in H_{0}^{1} (Ω), \forall n \geq 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).

Received: 2017-06-01
Accepted: 2017-06-07
Published online: 2017-06-27

DOI: 10.1016/j.crma.2017.06.004

Author's affiliations:

Haïm Brezis ^{1, 2, 3}; David Gómez-Castro ^{4, 5}

¹ Department of Mathematics, Hill Center, Busch Campus, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA
² Departments of Mathematics and Computer Science, Technion, Israel Institute of Technology, 32000 Haifa, Israel
³ Laboratoire Jacques-Louis-Lions, Université Pierre-et-Marie-Curie, 4, place Jussieu, 75252 Paris cedex 05, France
⁴ Dpto. de Matemática Aplicada, Universidad Complutense de Madrid, Spain
⁵ Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, Spain

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     author = {Ha{\"\i}m Brezis and David G\'omez-Castro},
     title = {Rigidity of optimal bases for signal spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {780--785},
     publisher = {Elsevier},
     volume = {355},
     number = {7},
     year = {2017},
     doi = {10.1016/j.crma.2017.06.004},
     language = {en},
}

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Haïm Brezis; David Gómez-Castro. Rigidity of optimal bases for signal spaces. Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 780-785. doi : 10.1016/j.crma.2017.06.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.06.004/

References
Cited by

[1] Y. Aflalo; H. Brezis; A. Bruckstein; R. Kimmel; N. Sochen Best bases for signal spaces, C. R. Acad. Sci. Paris, Ser. I, Volume 354 (2016) no. 12, pp. 1155-1167

[2] Y. Aflalo; H. Brezis; R. Kimmel On the optimality of shape and data representation in the spectral domain, SIAM J. Imaging Sci., Volume 8 (2015) no. 2, pp. 1141-1160

[3] R. Courant Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik, Math. Z., Volume 7 (1920) no. 1–4, pp. 1-57

[4] P.D. Lax Functional Analysis, John Wiley & Sons, New York, Chichester, 2002

[5] H. Poincaré Sur les équations aux dérivées partielles de la physique mathématique, Amer. J. Math., Volume 12 (1890) no. 3, pp. 211-294

[6] H.F. Weinberger Variational Methods for Eigenvalue Approximation, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1974

[7] H. Weyl Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann., Volume 71 (1912), pp. 441-479

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