  We present a unified approach for constructing Slepian functions - also known
as prolate spheroidal wave functions - on the sphere for arbitrary tensor ranks
including scalar, vectorial, and rank 2 tensorial Slepian functions, using
spin-weighted spherical harmonics. For the special case of spherical cap
regions, we derived commuting operators, allowing for a numerically stable and
computationally efficient construction of the spin-weighted
spherical-harmonic-based Slepian functions. Linear relationships between the
spin-weighted and the classical scalar, vectorial, tensorial, and higher-rank
spherical harmonics allow the construction of classical
spherical-harmonic-based Slepian functions from their spin-weighted
counterparts, effectively rendering the construction of spherical-cap Slepian
functions for any tensorial rank a computationally fast and numerically stable
task.
