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

2021/03/26 by Volker Michel, Alain Plattner, Michel, Volker +3
Earth and Planetary Sciences · Mathematics · #33C55 #41A10 #41A63 #42C05 #42C10 #42C25 #43A90 #45C05 #86-08 #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Geophysics and Gravity Measurements #Mathematical Physics (math-ph) #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2103.14650

openalex publication_date 2021/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

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