Iterative method to compute the maximal concentration Slepian band-limited eigenfunction on the sphere

The Slepian concentration problem on the sphere to maximize the energy concentration of a band-limited (in spherical harmonic degree) function is formulated as an eigenvalue problem, the solution of which gives rise to a family of band-limited eigenfunctions with optimal energy concentration in the spatial region on the sphere. In the family of band-limited eigenfunctions, the most concentrated (in the spatial region) eigenfunction is used for spatial windowing and spatial smoothing. We develop an iterative method to accurately compute the most concentrated band-limited Slepian eigenfunction for a given band-limit and a spatial region of interest. Taking into account the computational issues around the proposed iterative method, we also present the procedure for the practical implementation of the proposed method. In comparison to the computation of most concentration eigenfunction by the eigenvalue decomposition which gives a family of eigenfunction, the proposed method is computationally feasible even for large band-limits. Through examples, we also show that the proposed method attains sufficient numerical accuracy.