Spatio-Spectral Concentration on the Sphere

Spatio-spectral concentration theory is concerned with the optimal space localization of a signal bandlimited in the frequency domain. The question was first considered in a series of classic papers by Slepian, Pollak, and Landau [1961; 1961; 1962; 1978] who studied the problem on the real line; an introduction to the early work on the subject can be found in two articles by Slepian [1976; 1983]. Recently, Simons and co-workers [2006; 2009; 2010b] extended these results to the sphere for functions bandlimited in the Spherical Harmonics domain. In the following, we will fix notation and then provide an introduction to the spatio-spectral concentration problem on the sphere. Our presentation will closely follow the exposition by Simons et al. [2006].