Uniform approximation and explicit estimates for the prolate spheroidal wave functions

TL;DR

This paper advances the understanding of Prolate Spheroidal Wave Functions by providing explicit uniform approximation bounds, value estimates at 1, and eigenvalue formulas, enhancing their practical applicability in band-limited function analysis.

Contribution

It introduces explicit uniform approximation error bounds and formulas for PSWF values and eigenvalues, improving their analytical and computational handling.

Findings

Derived explicit uniform approximation error bounds for PSWFs.

Provided an explicit approximation of PSWF values at 1 using elliptic integrals.

Formulated an accurate approximation for the eigenvalues of the associated Sturm-Liouville operator.

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.