  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.
