Asymptotics of Prolate Spheroidal Wave Functions

TL;DR

This paper derives uniform asymptotic approximations for prolate spheroidal wave functions in high-frequency regimes, connecting parameters and providing error bounds using classical special functions.

Contribution

It introduces new uniform asymptotic formulas for prolate spheroidal wave functions with explicit error bounds, extending existing differential equation solutions.

Findings

Asymptotic formulas involve elementary, Bessel, and parabolic cylinder functions.

An asymptotic relationship between parameters is established.

Error bounds are provided for all approximations.

Abstract

Uniform asymptotic approximations are obtained for the prolate spheroidal wave functions, in the high-frequency case. The results are obtained by an application of certain existing asymptotic solutions of differential equations, and involve elementary, Bessel, and parabolic cylinder functions. An asymptotic relationship between the prolate spheroidal equation separation parameter and the other parameters is also obtained, and error bounds are available for all approximations