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We propose a novel basis of vector functions, the mixed vector spherical harmonics that are closely related to the functions of Sheppard and Török and help us reduce the concentration problem of tangential vector fields within a spherical cap to an equivalent scalar problem. Exploiting an analogy with previous results published by Grünbaum, Longhi and Perlstadt, we construct a differential operator that commutes with the concentration operator of this scalar problem and propose a stable and convenient method to obtain its eigenfunctions. Having obtained the scalar eigenfunctions, the calculation of tangential vector Slepian functions is straightforward.

Revisiting the Concentration Problem of Vector Fields within a Spherical Cap: A Commuting Differential Operator Solution