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We present a unified approach for constructing Slepian functions - also knownas prolate spheroidal wave functions - on the sphere for arbitrary tensor ranksincluding scalar, vectorial, and rank 2 tensorial Slepian functions, usingspin-weighted spherical harmonics. For the special case of spherical capregions, we derived commuting operators, allowing for a numerically stable andcomputationally efficient construction of the spin-weightedspherical-harmonic-based Slepian functions. Linear relationships between thespin-weighted and the classical scalar, vectorial, tensorial, and higher-rankspherical harmonics allow the construction of classicalspherical-harmonic-based Slepian functions from their spin-weightedcounterparts, effectively rendering the construction of spherical-cap Slepianfunctions for any tensorial rank a computationally fast and numerically stabletask.

A Unified Approach to Scalar, Vector, and Tensor Slepian Functions on the Sphere and Their Construction by a Commuting Operator