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 [3, 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 nonpolynomial basis. We in particular highlight that the collocation scheme together with a very practical rule for pairing up (c, N ) significantly outperforms the Legendre polynomialbased method (and likewise other Jacobi polynomial-based method) in approximating highly oscillatory bandlimited functions.1991 Mathematics Subject Classification. 65N35, 65E05, 65M70, 41A05, 41A10, 41A25.

This paper is devoted to a review of the prolate spheroidal wave functions (PSWFs) and their variants from the viewpoint of spectral/spectral-element approximations using such functions as basis functions. We demonstrate the pros and cons over their polynomial counterparts, and put the emphasis on the construction of essential building blocks for efficient spectral algorithms.

Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decay very slowly, subject to certain power laws. Their numerical solutions are under-explored. This paper aims at developing accurate spectral methods using rational basis (or modified mapped Gegenbauer functions) for such models in unbounded domains. The main building block of the spectral algorithms is the explicit representations for the Fourier transform and fractional Laplacian of the rational basis, derived from some useful integral identites related to modified Bessel functions. With these at our disposal, we can construct rational spectral-Galerkin and direct collocation schemes by pre-computing the associated fractional differentiation matrices. We obtain optimal error estimates of rational spectral approximation in the fractional Sobolev spaces, and analyze the optimal convergence of the proposed Galerkin scheme. We also provide ample numerical results to show that the rational method outperforms the Hermite function approach.

In this paper, a nonpolynomial-based spectral collocation method and its wellconditioned variant are proposed and analyzed. First, we develop fractional differentiation matrices of nodal Jacobi polyfractonomials [M.

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