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Uniform Approximation and Explicit Estimates for the Prolate Spheroidal Wave Functions

Abstract : For xed 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 largely 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 rst kind. Also, we give an explicit formula for the accurate approximation the eigenvalues of the Sturm-Liouville operator associated with the PSWFs. 2010 Mathematics Subject Classication. Primary 42C10, 65L70. Secondary 41A60, 65L15.
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Aline Bonami, Abderrazek Karoui. Uniform Approximation and Explicit Estimates for the Prolate Spheroidal Wave Functions. Constructive Approximation, Springer Verlag, 2015, ⟨10.1007/s00365-015-9295-1⟩. ⟨hal-01202320⟩

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