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

TL;DR

This paper introduces a unified, efficient method for constructing Slepian functions of various tensor ranks on the sphere, utilizing spin-weighted spherical harmonics and commuting operators for spherical cap regions.

Contribution

It develops a unified framework for scalar, vector, and tensor Slepian functions on the sphere, enabling fast and stable construction via commuting operators and linear relationships.

Findings

Derived commuting operators for spherical cap regions.

Established linear relationships between spin-weighted and classical spherical harmonics.

Enabled computationally efficient construction of tensorial Slepian functions.

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.