Highly Accurate Pseudospectral Approximations of the Prolate Spheroidal Wave Equation for Any Bandwidth Parameter and Zonal Wavenumber

A Review of Prolate Spheroidal Wave Functions from the Perspective of Spectral Methods

“…We refer to [96, Appendix A] for the explicit formulas for the entries of D (1) and D (2) , which only involve the function values {ψ N (x j )} N j=0 . In what follows, let D (2) in ∈ R (N−1)×(N−1) be the matrix obtained by deleting the first and last rows and columns of D (2) , and likewise for D (2) in .…”

“…In this section, we review the approach in [96] for computing the approximate "inverse" of D (2) in or D (2) in in a stable manner. With this, we can precondition the prolate-collocation method, leading to well-conditioned collocation schemes.…”

“…We essentially extend the idea for polynomials in [19,34,93] to the PSWFs. That is, we look for a matrix 35) for large N, and likewise for D (2) in . Such a matrix is generated from a new basis obtained from the generalised Birkhoff interpolation.…”

“…Note that the entries of E (2) are the second derivative of prolate-interpolation errors of {B k } (which are analytic functions). It is expected that they decay exponentially when N is large.…”

“…The above approach also applies to the cardinal basis {l j } and the corresponding prolate-differentiation matrix D (2) in .…”