SpatioSpectral Concentration on the Sphere
Christian Lessig and Eugene Fiume















Spherical
Slepian functions for L = 20 and a spherical cap
with θ ≤ 15 degree (dotted circle). The first row shows the first five
Slepian functions corresponding to the five largest eigenvalues and the
second row Slepian functions with the corresponding eigenvalues very
close to zero. Positive values are shown in blue and negative values in
red. Very small values are left white.

Abstract: Spatiospectral
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 coworkers
[2006; 2009; 2010b] extended these results to the sphere. We present a
concise and selfcontained introduction to spatiospectral
concentration theory.
See also our paper on the effective dimension of light transport in a local neighborhood where we use Slepian functions.
Technical Report
Christian Lessig and Eugene Fiume, SpatioSpectral
Concentration on the Sphere, Technical Report, University of
Toronto. (preprint)
Miscellaneous
Source code will be available shortly. Please send us a note if you
need it urgently.
Acknowledgement
We thank Tyler de Witt and Jos Stam for helpful
discussion. The implementation of Slepian functions by Frederik Simons was also valuable for our research.
