New efficient methods of computing the prolate spheroidal wave functions and their corresponding eigenvalues

In this paper, we describe different methods of computing the eigenvalues associated with the prolate spheroidal wave functions (PSWFs). These eigenvalues play an important role in computing the values of PSWFs as well as in the different numerical applications based on these later. The methods given in this work are accurate, fast and valid for small as well as for large values of the bandwidth c of the PSWFs. Moreover, we provide the reader with a method for computing the exact values of PSWFs at the Shannon sampling points. A Shannon sampling theorem with a better decaying sampling basis functions is used to provide a standard representation of the PSWFs. Moreover, we provide the reader with a new fast and accurate method for computing the PSFWFs which is valid over the real line. Based on this method, we develop asymptotic expansions of the PSWFs. Some numerical examples are given to illustrate the results of this paper.