笛卡尔平面上的空间谱浓度
Spatiospectral concentration in the Cartesian plane
AI总结：
针对自然科学应用，提出并解决了二维平面上的Slepian时频浓度问题，通过求解Fredholm积分方程得到正交的严格带限函数族，这些函数在给定区域内具有最优的空间谱浓度。
摘要：
我们提出并解决了二维平面上的Slepian时频浓度问题的类似问题，用于自然科学中的应用。我们确定了一个正交的严格带限函数族，这些函数在平面的一个封闭区域内最优地集中，或者，严格空间限制的函数在傅里叶域中最优地集中。笛卡尔Slepian函数可以通过求解Fredholm积分方程得到，其相关特征值是空间谱浓度的度量。原则上，空间和谱集中区域可以具有任意几何形状。然而，对于实际应用，如地球物理学或天文学中的信号表示或谱分析，物理空间中通常偏好不规则形状，而谱空间中通常偏好对称域。当集中域在两个空间中都是圆对称时，Slepian函数也是Sturm-Liouville算子的特征函数，这导致这种情况下的特殊算法，这是众所周知的。与它们的一维和球面对应物（我们在一个共同框架中讨论它们）非常相似，在任意笛卡尔域上同时具有空间和谱局部化的函数基将在许多科学学科中非常有用，尤其是在地球科学中。
英文摘要：
We pose and solve the analogue of Slepian's time-frequency concentration problem in the two-dimensional plane, for applications in the natural sciences. We determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the plane, or, alternatively, of strictly spacelimited functions that are optimally concentrated in the Fourier domain. The Cartesian Slepian functions can be found by solving a Fredholm integral equation whose associated eigenvalues are a measure of the spatiospectral concentration. Both the spatial and spectral regions of concentration can, in principle, have arbitrary geometry. However, for practical applications of signal representation or spectral analysis such as exist in geophysics or astronomy, in physical space irregular shapes, and in spectral space symmetric domains will usually be preferred. When the concentration domains are circularly symmetric in both spaces, the Slepian functions are also eigenfunctions of a Sturm-Liouville operator, leading to special algorithms for this case, as is well known. Much like their one-dimensional and spherical counterparts with which we discuss them in a common framework, a basis of functions that are simultaneously spatially and spectrally localized on arbitrary Cartesian domains will be of great utility in many scientific disciplines, but especially in the geosciences.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
