  We formulate and solve the Slepian spatial-spectral concentration problem on
the three-dimensional ball. Both the standard Fourier-Bessel and also the
Fourier-Laguerre spectral domains are considered since the latter exhibits a
number of practical advantages (spectral decoupling and exact computation). The
Slepian spatial and spectral concentration problems are formulated as
eigenvalue problems, the eigenfunctions of which form an orthogonal family of
concentrated functions. Equivalence between the spatial and spectral problems
is shown. The spherical Shannon number on the ball is derived, which acts as
the analog of the space-bandwidth product in the Euclidean setting, giving an
estimate of the number of concentrated eigenfunctions and thus the dimension of
the space of functions that can be concentrated in both the spatial and
spectral domains simultaneously. Various symmetries of the spatial region are
considered that reduce considerably the computational burden of recovering
eigenfunctions, either by decoupling the problem into smaller subproblems or by
affording analytic calculations. The family of concentrated eigenfunctions
forms a Slepian basis that can be used be represent concentrated signals
efficiently. We illustrate our results with numerical examples and show that
the Slepian basis indeeds permits a sparse representation of concentrated
signals.
