  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.
