Asymptotic behaviors and numerical computations of the eigenfunctions and eigenvalues associated with the classical and circular prolate spheroidal wave functions

For a fixed bandwidth c, the classical prolate spheroidal wave functions (PSWFs) ψn,c, and the circular ones (CPSWFs) φn,c, form remarkable Hilbertian bases for the spaces of Fourier and Hankel band-limited functions with bandwidth c, respectively. The prolate spheroidal wave functions have already found many applications from various scientific area such as signal processing, numerical analysis, Physics and random matrix theory. These concrete applications of the PSWFs and CPSWFs, require the accurate computation of these special functions. To this end, we develop in this work two approximate procedures. The first procedure is based on pushing forward the WKB method and provide accurate approximations of the PSWFs and the CPSWFs in terms of some classical special functions. The second procedure is based on efficient and accurate quadrature methods for the computation of the values and the eigenvalues associated with PSWFs and the CPSWFs. Both procedures are valid for small as well as for large values of the frequency band-width c. Also, we provide the reader with some numerical examples that illustrate the different results of this work.