Uniform approximation and explicit estimates for the prolate spheroidal wave functions

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 largely 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 the eigenvalues of the Sturm-Liouville operator associated with the PSWFs.

References in corpus (1)

• Spectral Decay of Time and Frequency Limiting Operator

Cited by in corpus (5)

• Spectral Decay of Time and Frequency Limiting Operator
• Non-Asymptotic behaviour of the spectrum of the Sinc Kernel Operator and Related Applications
• Generalized Prolate Spheroidal Wave Functions: Spectral Analysis and Approximation of Almost Band-limited Functions
• Weighted finite Fourier transform operator: Uniform approximations of the eigenfunctions, eigenvalues decay and behaviour
• The CMV bispectral problem