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Comptes Rendus Mathématique
Volume 352, n° 3
pages 229-234 (mars 2014)
Doi : 10.1016/j.crma.2014.01.004
Received : 27 July 2013 ;  accepted : 13 January 2014
Uniform bounds of prolate spheroidal wave functions and eigenvalues decay
Bornes uniformes des fonctions d'ondes sphéroidales et décroissance des valeurs propres
 

Aline Bonami a , Abderrazek Karoui b
a MAPMO–UMR 7349, CNRS–Université d'Orléans, 45067 Orléans, France 
b University of Carthage, Department of Mathematics, Faculty of Sciences of Bizerte, Jarzouna, 7021, Tunisia 

Abstract

The prolate spheroidal wave functions (PSWFs) form a set of special functions with remarkable properties. They are defined on   as the bounded eigenfunctions   of a Sturm–Liouville differential operator   as well as the eigenfunctions of the linear integral operator   with kernel  . We give new bounds for the values  ,   and  , which allow us to obtain estimates for the   norms of the PSWFs and for eigenvalues of   and  . We get in particular an almost sharp exponential lower decay rate of the eigenvalues of  .

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Résumé

Les fonctions d'ondes sphéroidales (PSWF) forment un ensemble de fonctions spéciales aux propriétés remarquables. Ces fonctions   sont à la fois les fonctions propres d'un opérateur différentiel   de type Sturm–Liouville sur   et de l'opérateur intégral   de noyau  . Nous donnons de nouvelles bornes pour les valeurs  ,   et  , ce qui nous permet d'obtenir des estimations des normes   des PSWF ainsi que des valeurs propres des opérateurs   et  . Nous donnons en particulier une borne inférieure presque critique des valeurs propres de l'opérateur  .

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 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.



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