This paper presents a novel approach for estimating the Koopman operator defined on a reproducing kernel Hilbert space (RKHS) and its spectra. We propose an estimation method, what we call Jet Dynamic Mode Decomposition (JetDMD), leveraging the intrinsic structure of RKHS and the geometric notion known as jets to enhance the estimation of the Koopman operator. This method refines the traditional Extended Dynamic Mode Decomposition (EDMD) in accuracy, especially in the numerical estimation of eigenvalues. This paper proves JetDMD's superiority through explicit error bounds and convergence rate for special positive definite kernels, offering a solid theoretical foundation for its performance. We also delve into the spectral analysis of the Koopman operator, proposing the notion of extended Koopman operator within a framework of rigged Hilbert space. This notion leads to a deeper understanding of estimated Koopman eigenfunctions and capturing them outside the original function space. Through the theory of rigged Hilbert space, our study provides a principled methodology to analyze the estimated spectrum and eigenfunctions of Koopman operators, and enables eigendecomposition within a rigged RKHS. We also propose a new effective method for reconstructing the dynamical system from temporally-sampled trajectory data of the dynamical system with solid theoretical guarantee. We conduct several numerical simulations using the van der Pol oscillator, the Duffing oscillator, the H\'enon map, and the Lorenz attractor, and illustrate the performance of JetDMD with clear numerical computations of eigenvalues and accurate predictions of the dynamical systems.

翻译：本文提出了一种在再生核希尔伯特空间（RKHS）上估计Koopman算子及其谱的新方法。我们提出了一种称为Jet动态模态分解（JetDMD）的估计方法，该方法利用RKHS的内在结构和称为jets的几何概念来增强Koopman算子的估计。该方法在精度上改进了传统的扩展动态模态分解（EDMD），特别是在特征值的数值估计方面。本文通过特殊正定核的显式误差界和收敛速度证明了JetDMD的优越性，为其性能提供了坚实的理论基础。我们还深入研究了Koopman算子的谱分析，在装备希尔伯特空间的框架下提出了扩展Koopman算子的概念。这一概念有助于更深入地理解估计的Koopman本征函数，并能够在原始函数空间之外捕获它们。通过装备希尔伯特空间理论，我们的研究为分析Koopman算子的估计谱和本征函数提供了一种原则性的方法论，并能够在装备RKHS中进行特征分解。我们还提出了一种新的有效方法，基于动态系统的时间采样轨迹数据重构动态系统，并具有坚实的理论保证。我们使用van der Pol振荡器、Duffing振荡器、Hénon映射和Lorenz吸引子进行了多次数值模拟，并通过清晰的特征值数值计算和动态系统的准确预测展示了JetDMD的性能。