球面上向量场的空间谱集中
Spatiospectral concentration of vector fields on a sphere
摘要：
我们构建的球面向量基地，是有限的和空间集中的，或者，空间有限的和频谱集中，适合于分析和表示的单位球的表面上的实值向量场，在自然和生物医学科学，工程。在Slepian，Landau和Pollak的原始方法的基础上，我们将函数基的能量集中到球体上的任意形状的感兴趣区域中，并且在矢量球谐域的某些带限内。与球上标量函数的浓度问题一样，Slepian向量基可以通过求解有限维代数本征值问题来构造。本征值问题分解为径向和切向分量的独立问题。对于具有高级对称性的区域，例如极冠，光谱浓度核矩阵非常容易计算和块对角化，从而使其本身有效对角化。空间频谱良好集中的向量场的数量很好地估计的香农数，只取决于目标区域的面积和最大球谐度或带宽。球面Slepian矢量基在整个球面和地理目标区域上都是双正交的。像它的标量对应物一样，它应该是一个强大的工具，用于反演，近似和扩展球上的带限场：向量场，如地球和行星科学中的重力和磁力，或光学，天线理论和医学成像中的电磁场。
英文摘要：
We construct spherical vector bases that are bandlimited and spatially concentrated, or, alternatively, spacelimited and spectrally concentrated, suitable for the analysis and representation of real-valued vector fields on the surface of the unit sphere, as arises in the natural and biomedical sciences, and engineering. Building on the original approach of Slepian, Landau, and Pollak we concentrate the energy of our function bases into arbitrarily shaped regions of interest on the sphere, and within certain bandlimits in the vector spherical-harmonic domain. As with the concentration problem for scalar functions on the sphere, which has been treated in detail elsewhere, a Slepian vector basis can be constructed by solving a finite-dimensional algebraic eigenvalue problem. The eigenvalue problem decouples into separate problems for the radial and tangential components. For regions with advanced symmetry such as polar caps, the spectral concentration kernel matrix is very easily calculated and block-diagonal, lending itself to efficient diagonalization. The number of spatiospectrally well-concentrated vector fields is well estimated by a Shannon number that only depends on the area of the target region and the maximal spherical-harmonic degree or bandwidth. The spherical Slepian vector basis is doubly orthogonal, both over the entire sphere and over the geographic target region. Like its scalar counterparts it should be a powerful tool in the inversion, approximation and extension of bandlimited fields on the sphere: vector fields such as gravity and magnetism in the earth and planetary sciences, or electromagnetic fields in optics, antenna theory and medical imaging.
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