  We introduce the Rigged Dynamic Mode Decomposition (Rigged DMD) algorithm,
which computes generalized eigenfunction decompositions of Koopman operators.
By considering the evolution of observables, Koopman operators transform
complex nonlinear dynamics into a linear framework suitable for spectral
analysis. While powerful, traditional Dynamic Mode Decomposition (DMD)
techniques often struggle with continuous spectra. Rigged DMD addresses these
challenges with a data-driven methodology that approximates the Koopman
operator's resolvent and its generalized eigenfunctions using snapshot data
from the system's evolution. At its core, Rigged DMD builds wave-packet
approximations for generalized Koopman eigenfunctions and modes by integrating
Measure-Preserving Extended Dynamic Mode Decomposition with high-order kernels
for smoothing. This provides a robust decomposition encompassing both discrete
and continuous spectral elements. We derive explicit high-order convergence
theorems for generalized eigenfunctions and spectral measures. Additionally, we
propose a novel framework for constructing rigged Hilbert spaces using
time-delay embedding, significantly extending the algorithm's applicability. We
provide examples, including systems with a Lebesgue spectrum, integrable
Hamiltonian systems, the Lorenz system, and a high-Reynolds number lid-driven
flow in a two-dimensional square cavity, demonstrating Rigged DMD's
convergence, efficiency, and versatility. This work paves the way for future
research and applications of decompositions with continuous spectra.
