Uniform Approximation and Explicit Estimates for the Prolate Spheroidal Wave Functions

For fixed c , prolate spheroidal wave functions (PSWFs), denoted by ψ 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 appli...

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Bibliographic Details
Published in:Constructive approximation 2016-02, Vol.43 (1), p.15-45
Main Authors: Bonami, Aline, Karoui, Abderrazek
Format: Article
Language:English
Subjects:
Analysis
Classical Analysis and ODEs
Mathematics
Mathematics and Statistics
Numerical Analysis
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Summary:For fixed c , prolate spheroidal wave functions (PSWFs), denoted by ψ 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 ψ 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.
ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-015-9295-1