A steepest ascent algorithm for the calculation of generalized prolate spheroidal wavefunctions

In this report, a steepest ascent algorithm for eigenpair calculation [1] is used to derive the eigenpairs of a 2D space-limited band-pass operator. The peculiar difficulty in this application is essentially due to the large size of the matrix of the discretized operator, and, for some cases, to the presence of several numerically degenerate eigenvalues. The eigenfunctions of the considered operator are related to the generalized prolate spheroidal wavefunctions [2].The problem of calculating the eigenpairs of a linear operator will be shown to be equivalent to searching for the stationary points of the Rayleigh quotient, and the algorithm will be described in detail. The theoretical considerations on the properties of the particular operator studied here will be carried out in a continuous setting; the details of the discretization of the problem will be described successively. The numerical results for some cases of interest will be then reported and an application to the study of a superresolution iterative algorithm will be shown.