Uniform bounds of prolate spheroidal wave functions and eigenvalues decay

Bonami, Aline; Karoui, Abderrazek

doi:10.1016/j.crma.2014.01.004

Harmonic analysis

Uniform bounds of prolate spheroidal wave functions and eigenvalues decay
[Bornes uniformes des fonctions d'ondes sphéroidales et décroissance des valeurs propres]

Présenté par : the Editorial Board

Bonami, Aline ¹ ; Karoui, Abderrazek ²

¹ MAPMO–UMR 7349, CNRS–Université d'Orléans, 45067 Orléans, France
² University of Carthage, Department of Mathematics, Faculty of Sciences of Bizerte, Jarzouna, 7021, Tunisia

Comptes Rendus. Mathématique, Tome 352 (2014) no. 3, pp. 229-234.

Résumé
Abstract

Les fonctions d'ondes sphéroidales (PSWF) forment un ensemble de fonctions spéciales aux propriétés remarquables. Ces fonctions $ψ_{n, c}$ sont à la fois les fonctions propres d'un opérateur différentiel $L_{c}$ de type Sturm–Liouville sur $[- 1, 1]$ et de l'opérateur intégral $Q_{c}$ de noyau $\frac{\sin (c (x - y))}{π (x - y)}$ . Nous donnons de nouvelles bornes pour les valeurs $ψ_{n, c} (0)$ , $ψ_{n, c}^{'} (0)$ et $ψ_{n, c} (1)$ , ce qui nous permet d'obtenir des estimations des normes $L^{p}$ des PSWF ainsi que des valeurs propres des opérateurs $L_{c}$ et $Q_{c}$ . Nous donnons en particulier une borne inférieure presque critique des valeurs propres de l'opérateur $Q_{c}$ .

The prolate spheroidal wave functions (PSWFs) form a set of special functions with remarkable properties. They are defined on $[- 1, 1]$ as the bounded eigenfunctions $ψ_{n, c}$ of a Sturm–Liouville differential operator $L_{c}$ as well as the eigenfunctions of the linear integral operator $Q_{c}$ with kernel $\frac{\sin (c (x - y))}{π (x - y)}$ . We give new bounds for the values $ψ_{n, c} (0)$ , $ψ_{n, c}^{'} (0)$ and $ψ_{n, c} (1)$ , which allow us to obtain estimates for the $L^{p}$ norms of the PSWFs and for eigenvalues of $L_{c}$ and $Q_{c}$ . We get in particular an almost sharp exponential lower decay rate of the eigenvalues of $Q_{c}$ .

Reçu le : 2013-07-27
Accepté le : 2014-01-13
Publié le : 2014-02-05

DOI : 10.1016/j.crma.2014.01.004

Affiliations des auteurs :

Bonami, Aline ¹ ; Karoui, Abderrazek ²

¹ MAPMO–UMR 7349, CNRS–Université d'Orléans, 45067 Orléans, France
² University of Carthage, Department of Mathematics, Faculty of Sciences of Bizerte, Jarzouna, 7021, Tunisia

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Bonami, Aline; Karoui, Abderrazek. Uniform bounds of prolate spheroidal wave functions and eigenvalues decay. Comptes Rendus. Mathématique, Tome 352 (2014) no. 3, pp. 229-234. doi : 10.1016/j.crma.2014.01.004. http://www.numdam.org/articles/10.1016/j.crma.2014.01.004/

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Cité par Sources :

^☆ This work was supported in part by the ANR grant “AHPI” ANR-07-BLAN-0247-01, the French–Tunisian CMCU 10G 1503 and the DGRST Research Grant 05 UR 15-02.