  In this paper, we develop a new method for the fast and memory-efficient
computation of Slepian functions on the sphere. Slepian functions, which arise
as the solution of the Slepian concentration problem on the sphere, have
desirable properties for applications where measurements are only available
within a spatially limited region on the sphere and/or a function is required
to be analyzed over the spatially limited region. Slepian functions are
currently not easily computed for large band-limits for an arbitrary spatial
region due to high computational and large memory storage requirements. For the
special case of a polar cap, the symmetry of the region enables the
decomposition of the Slepian concentration problem into smaller subproblems and
consequently the efficient computation of Slepian functions for large
band-limits. By exploiting the efficient computation of Slepian functions for
the polar cap region on the sphere, we develop a formulation, supported by a
fast algorithm, for the approximate computation of Slepian functions for an
arbitrary spatial region to enable the analysis of modern datasets that support
large band-limits. For the proposed algorithm, we carry out accuracy analysis
of the approximation, computational complexity analysis, and review of memory
storage requirements. We illustrate, through numerical experiments, that the
proposed method enables faster computation, and has smaller storage
requirements, while allowing for sufficiently accurate computation of the
Slepian functions.
