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arXiv · 1302.5261

Revisiting the concentration problem of vector fields within a spherical cap: a commuting differential operator solution

Kornél Jahn·Nándor Bokor
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Abstract

We propose a novel basis of vector functions, the mixed vector spherical harmonics that are closely related to the functions $F_{lm}$ of Sheppard and T\"or\"ok 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 Gr\"unbaum and his colleagues, 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.

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Kornél Jahn, Nándor Bokor. 2013-02-21. Revisiting the concentration problem of vector fields within a spherical cap: a commuting differential operator solution. https://arxiv.org/abs/1302.5261

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