球面上标量、向量和张量Slepian函数的统一方法及其交换算子的构造
A Unified Approach to Scalar, Vector, and Tensor Slepian Functions on the Sphere and Their Construction by a Commuting Operator
摘要：
我们提出了一个统一的方法来构建Slepian函数-也被称为长球体波函数-在球体上的任意张量秩，包括标量，矢量和秩2张量Slepian函数，使用自旋加权球谐函数。对于特殊情况下的球冠区域，我们推导出交换算子，允许一个数值稳定和计算效率的自旋加权球谐基Slepian函数的建设。自旋加权和经典标量，矢量，张量和高阶球谐函数之间的线性关系允许从它们的自旋加权对应物构建经典的基于球谐的Slepian函数，有效地使任何张量秩的球帽Slepian函数的构建成为计算快速和数值稳定的任务。
英文摘要：
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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