  We construct spherical vector bases that are bandlimited and spatially
concentrated, or, alternatively, spacelimited and spectrally concentrated,
suitable for the analysis and representation of real-valued vector fields on
the surface of the unit sphere, as arises in the natural and biomedical
sciences, and engineering. Building on the original approach of Slepian,
Landau, and Pollak we concentrate the energy of our function bases into
arbitrarily shaped regions of interest on the sphere, and within certain
bandlimits in the vector spherical-harmonic domain. As with the concentration
problem for scalar functions on the sphere, which has been treated in detail
elsewhere, a Slepian vector basis can be constructed by solving a
finite-dimensional algebraic eigenvalue problem. The eigenvalue problem
decouples into separate problems for the radial and tangential components. For
regions with advanced symmetry such as polar caps, the spectral concentration
kernel matrix is very easily calculated and block-diagonal, lending itself to
efficient diagonalization. The number of spatiospectrally well-concentrated
vector fields is well estimated by a Shannon number that only depends on the
area of the target region and the maximal spherical-harmonic degree or
bandwidth. The spherical Slepian vector basis is doubly orthogonal, both over
the entire sphere and over the geographic target region. Like its scalar
counterparts it should be a powerful tool in the inversion, approximation and
extension of bandlimited fields on the sphere: vector fields such as gravity
and magnetism in the earth and planetary sciences, or electromagnetic fields in
optics, antenna theory and medical imaging.
