In this paper we generalize the spectral concentration problem as formulated by Slepian, Pollak and Landau in the 1960s. We show that a generalized version with arbitrary space and Fourier masks is well-posed, and we prove some new results concerning general quadratic domains and gaussian filters. We also propose a more general splitting representation of the spectral concentration operator allowing to construct quasi-modes in some situations. We then study its discretization and we illustrate the fact that standard eigen-algorithms are not robust because of a clustering of eigenvalues. We propose a new alternative algorithm that can be implemented in any dimension and for any domain shape, and that gives very efficient results in practice.

翻译：本文推广了Slepian、Pollak和Landau在20世纪60年代提出的谱集中问题。我们证明具有任意空间掩模与傅里叶掩模的广义版本是适定的，并针对一般二次型域和高斯滤波器证明了若干新结论。我们还提出了谱集中算子的广义分裂表示法，可在特定情形下构造拟模态。随后我们研究其离散化过程，并通过特征值聚集现象说明传统特征算法缺乏鲁棒性。我们提出一种新的替代算法，该算法可在任意维度与任意域形状中实现，并在实际应用中取得显著效果。