37 research outputs found
Uniform approximation and explicit estimates for the prolate spheroidal wave functions
For fixed Prolate Spheroidal Wave Functions (PSWFs), denoted by
form an orthogonal basis with remarkable properties for the
space of band-limited functions with bandwith . They have been largely
studied and used after the seminal work of D. Slepian and his co-authors. In
several applications, uniform estimates of the in and are
needed. To progress in this direction, we push forward the uniform
approximation error bounds and give an explicit approximation of their values
at in terms of the
Legendre complete elliptic integral of the first kind. Also, we give an
explicit formula for the accurate approximation the eigenvalues of the
Sturm-Liouville operator associated with the PSWFs
Spectral Decay of Time and Frequency Limiting Operator
For fixed the Prolate Spheroidal Wave Functions (PSWFs)
form a basis with remarkable properties for the space of band-limited functions
with bandwidth . They have been largely studied and used after the seminal
work of D. Slepian, H. Landau and H. Pollack. Many of the PSWFs applications
rely heavily of the behavior and the decay rate of the eigenvalues
of the time and frequency limiting operator, which
we denote by Hence, the issue of the accurate estimation of the
spectrum of this operator has attracted a considerable interest, both in
numerical and theoretical studies. In this work, we give an explicit integral
approximation formula for these eigenvalues. This approximation holds true
starting from the plunge region where the spectrum of starts to
have a fast decay. As a consequence of our explicit approximation formula, we
give a precise description of the super-exponential decay rate of the
Also, we mention that the described approximation scheme
provides us with fairly accurate approximations of the with low
computational load, even for very large values of the parameters and
Finally, we provide the reader with some numerical examples that illustrate the
different results of this work.Comment: arXiv admin note: substantial text overlap with arXiv:1012.388
Analyse harmonique et fonctions d'ondes sphéroïdales
Notre travail est motivé par le problème de l'évaluation du déterminant de Fredholm d'un opérateur intégral. Cet opérateur apparait dans l'expression de la probabilité pour qu'un intervalle [?s, s] (s > 0) ne contienne aucune valeur propre d'une matrice aléatoire hermitienne gaussienne. Cet opérateur commute avec un opérateur différentiel de second ordre dont les fonctions propres sont les fonctions d'ondes sphéroïdales de l'ellipsoïde alongé. Plus généralement nous considérons l'opérateur de Legendre perturbé. Nous montrons qu'il existe un opérateur de translation généralisée associé à cet opérateur. En?n, par une méthode d'approximation des solutions de certaines équations différentielles, dite méthode WKB, nous avons obtenu le comportement asymptotique des fonctions d'ondes sphéroïdales de l'ellipsoïde alongé Il s'exprime à l'aide des fonctions de Bessel et d'Airy. Par la même méthode nous avons obtenu le comportement asymptotique des fonctions propres de l'opérateur dfférentiel d'Airy.Our work is motivated by the problem of evaluating the Fredholm determinant of an integral operator. This operator appears in the expression of the probability, for a random matrix in the Gaussien Unitary Ensemble, to have no eigenvalue in an interval [?s, s]. This operator commutes with a differential operator wich have the spheroidal wave functions as eingenfunctions. More generally, we consider the perturbated Legendre differential operator. We show that there exists a generalized translation operator associated to the perturbated Legendre dfferential operator. Finaly, by using the WKB method, we have determined the asymptotic behavior of the prolate spheroidal wave functions. This asymptotic behavior involves Bessel and Airy functions. By using the same method, we have obtained similar results for asymptotic behavior of the eigenfunctions of the Airy differential operator.PARIS-JUSSIEU-Bib.électronique (751059901) / SudocSudocFranceF