A unified approach to scalar, vector, and tensor Slepian functions on the sphere and their construction by a commuting operator

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