A new generalization of the PSWFs with applications to spectral approximations on quasi-uniform grids

We define a new family of generalized prolate spheroidal wave functions (GPSWFs), which extends the prolate spheroidal wave functions of order zero (PSWFs or Slepian functions; Slepian and Pollak, 1961 [45]) to real order α>−1, and also generalizes the Gegenbauer polynomials to an orthogonal system with an intrinsic tuning parameter c>0. We show that the GPSWFs, defined as the eigenfunctions of a Sturm–Liouville problem, are also the eigenfunctions of an integral operator. We present a number of analytic and asymptotic formulae for the GPSWFs and the associated eigenvalues, and introduce efficient algorithms for their evaluations. Moreover, we derive a set of optimal results on the GPSWF approximations featured with explicit dependence on the parameter c. As an important application, we implement and analyze the GPSWF spectral methods for elliptic-type equations. We illustrate that the presence of c provides flexibility to design high-order approximations on quasi-uniform grids, and endows the GPSWFs with some favorable advantages over their polynomial counterparts.