%0 Journal Article
%T On $hp$-Convergence of PSWFs and A New Well-Conditioned Prolate-Collocation Scheme
%A Li-Lian Wang
%A Jing Zhang
%A Zhimin Zhang
%J Mathematics
%D 2013
%I arXiv
%R 10.1016/j.jcp.2014.03.005
%X The first purpose of this paper is to provide a rigorous proof for the nonconvergence of $h$-refinement in $hp$-approximation by the PSWFs, a surprising convergence property that was first observed by Boyd et al [J. Sci. Comput., 2013]. The second purpose is to offer a new basis that leads to spectral-collocation systems with condition numbers independent of $(c,N),$ the intrinsic bandwidth parameter and the number of collocation points. In addition, this work gives insights into the development of effective spectral algorithms using this non-polynomial basis. We in particular highlight that the collocation scheme together with a very practical rule for pairing up $(c,N)$ significantly outperforms the Legendre polynomial-based method (and likewise other Jacobi polynomial-based method) in approximating highly oscillatory bandlimited functions.
%U http://arxiv.org/abs/1310.3457v1