We pose and solve the analogue of Slepian's time-frequency
concentration problem on the surface of the unit sphere, to
determine an orthogonal family of strictly bandlimited functions
that are optimally concentrated within a closed region of the sphere,
or, alternatively, of strictly spacelimited functions that are
optimally concentrated within the spherical harmonic domain. Such a
basis of simultaneously spatially and spectrally concentrated
functions should be a useful data analysis and representation tool in
a variety of geophysical and planetary applications, as well as in
medical imaging, computer science, cosmology and numerical analysis.
The spherical Slepian functions can be found either by solving an
algebraic eigenvalue problem in the spectral domain or by solving a
Fredholm integral equation in the spatial domain. The associated
eigenvalues are a measure of the spatiospectral concentration. When
the concentration region is an axisymmetric polar cap the
spatiospectral projection operator commutes with a Sturm-Liouville
operator; this enables the eigenfunctions to be computed extremely
accurately and efficiently, even when their area-bandwidth product, or
Shannon number, is large. In the asymptotic limit of a small spatial
region and a large spherical harmonic bandwidth, the spherical
concentration problem reduces to its planar equivalent, which exhibits
self-similarity when the Shannon number is kept invariant.
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