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Constructive Approximation

, Volume 43, Issue 1, pp 15–45 | Cite as

Uniform Approximation and Explicit Estimates for the Prolate Spheroidal Wave Functions

  • Aline BonamiEmail author
  • Abderrazek Karoui
Article

Abstract

For fixed c, prolate spheroidal wave functions (PSWFs), denoted by \(\psi _{n, c}\), form an orthogonal basis with remarkable properties for the space of band-limited functions with bandwith c. They have been widely studied and used after the seminal work of D. Slepian and his co-authors. In several applications, uniform estimates of the \(\psi _{n,c}\) in n and c are needed. To progress in this direction, we push forward the uniform approximation error bounds and give an explicit approximation of their values at 1 in terms of the Legendre complete elliptic integral of the first kind. Also, we give an explicit formula for the accurate approximation of the eigenvalues of the Sturm–Liouville operator associated with the PSWFs.

Keywords

Prolate spheroidal wave functions Asymptotic and uniform estimates Eigenvalues and eigenfunctions Sturm–Liouville operator 

Mathematics Subject Classification

Primary 42C10 65L70 Secondary 41A60 65L15 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Fédération Denis-Poisson, MAPMO-UMR 7349, Département de MathématiquesUniversité d’OrléansOrléans Cedex 2France
  2. 2.Department of Mathematics, Faculty of Sciences of BizerteUniversity of CarthageTunisTunisia

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