A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. This framework is based on an approximation of the generator of the system by a compact operator $W_{\tau }$ on a reproducing kernel Hilbert space (RKHS). The operator $W_{\tau }$ is skew-adjoint, and thus can be represented by a projection-valued measure, discrete by compactness, with an associated orthonormal basis of eigenfunctions. These eigenfunctions are ordered in terms of a Dirichlet energy, and provide a notion of coherent observables under the dynamics akin to the Koopman eigenfunctions associated with the atomic part of the spectrum. In addition, $W_{\tau }$ generates a unitary evolution group ${\{e^{t{W}_{\tau }}\}}_{t\in \mathbb{R}}$ on the RKHS, which approximates the unitary Koopman group of the system. We establish convergence results for the spectrum and Borel functional calculus of $W_{\tau }$ as $\tau \to 0^{+}$, as well as an associated data-driven formulation utilizing time series data. Numerical applications to ergodic systems with atomic and continuous spectra, namely a torus rotation, the Lorenz 63 system, and the Rössler system, are presented.

建立了一个框架的框架，用于相干模式提取和具有原子和连续光谱成分的保全遍历动力学系统的可观观测值的预测。该框架基于紧凑型算子对系统生成器的近似值${\mathrm{w\; ^}}_{\tau }$在可再生内核希尔伯特空间（RKHS）上。运营商${\mathrm{w\; ^}}_{\tau }$是倾斜伴生的，因此可以用投影值度量来表示，该度量因紧致性而离散，并且具有本征函数的相关正交正交基础。这些本征函数根据狄利克雷能量进行排序，并在类似于与光谱的原子部分相关的库普曼本征函数的动力学下提供了相干可观测的概念。此外，${\mathrm{w\; ^}}_{\tau }$ 产生一个单一的进化组 ${\{{Ë}^{Ť{\mathrm{w\; ^}}_{\tau }}\}}_{Ť\in \mathbb{[R}}$在RKHS上，它近似于系统的单一Koopman组。我们为谱和Borel函数演算建立收敛结果${\mathrm{w\; ^}}_{\tau }$ 作为 $\tau \to 0^{+}$，以及使用时间序列数据的相关数据驱动公式。介绍了具有原子和连续光谱的遍历系统的数值应用，即圆环旋转，Lorenz 63系统和Rössler系统。