Starting from measured data, we develop a method to compute the fine structure of the spectrum of the Koopman operator with rigorous convergence guarantees. The method is based on the observation that, in the measure-preserving ergodic setting, the moments of the spectral measure associated to a given observable are computable from a single trajectory of this observable. Having finitely many moments available, we use the classical Christoffel–Darboux kernel to separate the atomic and absolutely continuous parts of the spectrum, supported by convergence guarantees as the number of moments tends to infinity. In addition, we propose a technique to detect the singular continuous part of the spectrum as well as two methods to approximate the spectral measure with guaranteed convergence in the weak topology, irrespective of whether the singular continuous part is present or not. The proposed method is simple to implement and readily applicable to large-scale systems since the computational complexity is dominated by inverting an $N\times N$ Hermitian positive-definite Toeplitz matrix, where N is the number of moments, for which efficient and numerically stable algorithms exist; in particular, the complexity of the approach is independent of the dimension of the underlying state-space. We also show how to compute, from measured data, the spectral projection on a given segment of the unit circle, allowing us to obtain a finite approximation of the operator that explicitly takes into account the point and continuous parts of the spectrum. Finally, we describe a relationship between the proposed method and the so-called Hankel Dynamic Mode Decomposition, providing new insights into the behavior of the eigenvalues of the Hankel DMD operator. A number of numerical examples illustrate the approach, including a study of the spectrum of the lid-driven two-dimensional cavity flow.

从实测数据开始，我们开发了一种在严格收敛保证下计算Koopman算子频谱精细结构的方法。该方法基于以下观察结果：在保留测度的遍历设置中，与给定可观测值相关的频谱量度的矩可以从该可观测值的单个轨迹计算出来。由于有有限的矩可用，我们使用经典的Christoffel–Darboux内核来分离频谱的原子部分和绝对连续部分，并通过收敛保证来保证，因为矩数趋于无穷大。此外，我们提出了一种检测频谱的奇异连续部分的技术，以及两种在弱拓扑中保证收敛的方法来近似频谱测量；不管是否存在奇异的连续部分。所提出的方法易于实现，并且容易适用于大规模系统，因为计算复杂度主要是通过对$ñ\times ñ$埃尔米特式正定Toeplitz矩阵，其中N是存在有效且数值稳定的算法的矩数；特别是，该方法的复杂性与基础状态空间的维数无关。我们还展示了如何从测量的数据计算在单位圆的给定段上的光谱投影，从而使我们能够获得算子的有限近似值，该近似值明确考虑了光谱的点和连续部分。最后，我们描述了所提出的方法与所谓的Hankel动态模式分解之间的关系，从而为Hankel DMD算子的特征值的行为提供了新的见解。大量的数值示例说明了这种方法，其中包括对盖子驱动的二维腔流的频谱进行研究。