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.
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Communicated by Erik Koelink.
This work was supported in part by the ANR Grant “AHPI” ANR-07-BLAN-0247-01, the French–Tunisian CMCU 10G 1503 Project and the DGRST research Grant UR 13 ES 47. Part of this work was done while the second author was visiting the research laboratory MAPMO of the University of Orléans, France.
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Bonami, A., Karoui, A. Uniform Approximation and Explicit Estimates for the Prolate Spheroidal Wave Functions. Constr Approx 43, 15–45 (2016). https://doi.org/10.1007/s00365-015-9295-1
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DOI: https://doi.org/10.1007/s00365-015-9295-1
Keywords
- Prolate spheroidal wave functions
- Asymptotic and uniform estimates
- Eigenvalues and eigenfunctions
- Sturm–Liouville operator