球面上的时空频谱集中
Spatiospectral concentration on a sphere
AI总结：
提出并求解球面上的Slepian时频集中问题，得到一组严格带限且最优集中在球面闭合区域内的正交函数，或严格空间受限且最优集中在球谐域内的函数。
摘要：
我们提出并求解了单位球面上的Slepian时频集中问题的类比，以确定一组严格带限且在球面闭合区域内最优集中的正交函数，或者严格空间受限且在球谐域内最优集中的函数。这种同时具有空间和频谱集中性质的函数基，在各种地球物理和行星应用、医学成像、计算机科学、宇宙学和数值分析中，应成为有用的数据分析和表示工具。球面Slepian函数可以通过在谱域中求解代数特征值问题或在空间域中求解Fredholm积分方程得到。相关的特征值是时空频谱集中的度量。当集中区域是轴对称极冠时，时空频谱投影算子与Sturm-Liouville算子交换；这使得即使面积-带宽乘积（或Shannon数）很大，也能极其精确和高效地计算特征函数。在小集中区域和大球谐带宽的渐近极限下，球面集中问题趋近于其平面等价形式，当Shannon数保持不变时，该等价形式表现出自相似性。
英文摘要：
We pose and solve the analogue of Slepian's time-frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere, or, alternatively, of strictly spacelimited functions that are optimally concentrated within the spherical harmonic domain. Such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications, as well as in medical imaging, computer science, cosmology and numerical analysis. The spherical Slepian functions can be found either by solving an algebraic eigenvalue problem in the spectral domain or by solving a Fredholm integral equation in the spatial domain. The associated eigenvalues are a measure of the spatiospectral concentration. When the concentration region is an axisymmetric polar cap the spatiospectral projection operator commutes with a Sturm-Liouville operator; this enables the eigenfunctions to be computed extremely accurately and efficiently, even when their area-bandwidth product, or Shannon number, is large. In the asymptotic limit of a small concentration region and a large spherical harmonic bandwidth the spherical concentration problem approaches its planar equivalent, which exhibits self-similarity when the Shannon number is kept invariant.
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