Differential and Weighted Slepian Concentration Problems on the Sphere

In this work, we present a generalized formulation of the Slepian concentration problem on the sphere for finding band-limited functions with an optimal concentration in the spatial domain. By introducing weighting functions in the formulation of classical Slepian concentration problem and assigning different values to these weighting functions, we present two variants of the concentration problem namely the differential and the weighted Slepian concentration problem. In the differential Slepian concentration problem, we consider two regions on the sphere and find band-limited functions such that the energy is maximized in one region at the expense of the energy in the other region. We propose non-negative weighting using a spatial window function to formulate and solve the weighted Slepian concentration problem. Each problem can be solved by formulating it in the harmonic domain as an eigenvalue problem, the solution of which yields eigenfunctions that serve as alternative basis functions for the representation of band-limited signals and are referred to as Slepian functions. We also present and analyse the properties of the Slepian functions. To support the applications in acoustics and cosmology, we also provide a demonstration for the use of the proposed Slepian functions for the robust signal modeling and the estimation of the energy spectrum of red and white stochastic processes on the sphere.