On hp-convergence of prolate spheroidal wave functions and a new well-conditioned prolate-collocation scheme

The first purpose of this paper is to provide further illustrations, from both theoretical and numerical perspectives, for the nonconvergence of h-refinement in hp-approximation by the prolate spheroidal wave functions (PSWFs), a surprising convergence property that was first discovered by Boyd et al. (2013) [3]. The second purpose is to offer a new basis that leads to prolate-collocation systems with condition numbers independent of (c,N)(c,N), the intrinsic bandwidth parameter and the number of collocation points. We highlight that the collocation scheme together with a very practical rule for pairing up (c,N)(c,N) significantly outperforms the Legendre polynomial-based method (and likewise other Jacobi polynomial-based methods) in approximating highly oscillatory bandlimited functions.