  In this paper, we develop an analytical formulation for the Slepian
spatial-spectral concentration problem on the sphere for a limited
colatitude-longitude spatial region on the sphere, defined as the Cartesian
product of a range of positive colatitudes and longitudes. The solution of the
Slepian problem is a set of functions that are optimally concentrated and
orthogonal within a spatial or spectral region. These properties make them
useful for applications where measurements are taken within a spatially limited
region of the sphere and/or a signal is only to be analyzed within a region of
the sphere. To support localized spectral/spatial analysis, and estimation and
sparse representation of localized data in these applications, we exploit the
expansion of spherical harmonics in the complex exponential basis to develop an
analytical formulation for the Slepian concentration problem for a limited
colatitude-longitude spatial region. We also extend the analytical formulation
for spatial regions that are comprised of a union of rotated limited
colatitude-longitude subregions. By exploiting various symmetries of the
proposed formulation, we design a computationally efficient algorithm for the
implementation of the proposed analytical formulation. Such a reduction in
computation time is demonstrated through numerical experiments. We present
illustrations of our results with the help of numerical examples and show that
the representation of a spatially concentrated signal is indeed sparse in the
Slepian basis.
