The Koopman Operator as a Discrete Fourier Transform for dynamical systems
Spectral analysis of the Koopman Operator:
Koopman modes:
Let’s suppose \(\vec{x} \in M \subset \mathbb{R}^n\) contains all the information concerning the flow field at a particular time, so \(g(x)\) may be a vector of any quantity of interest such as velocity measurements.
If we let \(\phi_j(x): M \rightarrow \mathbb{R}\) denote eigenfunctions and \(\lambda_j \in \mathbb{C}\) denote eigenvalues of the Koopman operator, we have:
\begin{equation} \forall j \in \mathbb{N}, U\phi_j(x) = \lambda_j \cdot \phi_j(x) \end{equation}
If each of the components of \(\vec{g}\) lie within the span of the eigenfunctions \(\phi_j\) then we may expand the vectorvalued \(\vec{g}\) in terms of the eigenfunctions:
\begin{equation} g(\vec{x}) = \sum_{j=1}^\infty \phi_j(x) \cdot \vec{v}_j \end{equation}
so we may think of \(\vec{g}\) as a linear combination of the eigenfunctions \(\phi_j\) of \(U\) where \(\vec{v}_j\) are the vector coefficients in the expansion.
By induction, we find the iterates of \(\vec{x}_0\) are given by:
\begin{equation} g(\vec{x_k}) = \sum_{j=1}^\infty U^k \circ \phi_j(\vec{x_0}) \cdot \vec{v_j} = \sum_{j=1}^\infty \lambda_j^k \cdot \phi_j(\vec{x_0}) \cdot \vec{v}_j \end{equation}
so we may think of the Koopman operator \(U\) as an infinite dimensional operator that acts on data from a Hilbert space.
The Koopman operator as a Discrete Fourier Transform for dynamical systems:
Building upon the notion that a Koopman operator may be decomposed into eigenfunctions we may show that in a large number of settings the Koopman operator performs an intelligent reparametrization of the data by discovering regularities(ex. periodic behaviour). To be precise, the Koopman operator may be identified with a Discrete Fourier Transform for dynamical systems.
For concreteness, if we have a dataset of vectors sampled from a periodic system \(\forall k \in \mathbb{N}, x_{k+m}=x_k\):
\begin{equation} D = \{\vec{x_0}, …, \vec{x_{m1}}\} \end{equation}
so we may compute its Discrete Fourier Transform:
\begin{equation} D \rightarrow DFT \rightarrow \hat{D} \end{equation}
\begin{equation} \forall j,k \in [0,m1], x_k = \sum_{j=0}^{m1} e^{\frac{2\pi ijk}{m}} \end{equation}
and so we may define a set of functions \(\phi_j:S \rightarrow \mathbb{C}\) by:
\begin{equation} \forall j,k \in [0,m1],\phi_j(\vec{x_k}) = e^{\frac{2\pi ijk}{m}} \end{equation}
Then \(\phi_j\) are eigenfunctions of the Koopman operator \(U\) with eigenvalues \(e^{\frac{2\pi ij}{m}}\) since:
\begin{equation} U\phi_j(x_k) = \phi_j \circ f(x_k) = \phi_j(x_{k+1}) = e^{\frac{2\pi ij(k+1)}{m}}= e^{\frac{2\pi ij}{m}}\phi_j(x_k) \end{equation}
and therefore \(\text{rank}(U)=m\), so we may express the expansion as:
\begin{equation} \vec{x_k} = \sum_{j=0}^{m1} \phi_j(x_k)\hat{x}_j \end{equation}
This result generalises to nonperiodic systems when the dynamics are restricted to an attractor.
References:

Bernard Koopman. Hamiltonian systems and Transformations in Hilbert Space.

Hassan Arbabi. Introduction to Koopman operator theory for dynamical systems. MIT. 2020.

Steven L. Brunton. Notes on Koopman operator theory. 2019.

C. Rowley et al. Spectral analysis of nonlinear flows.

Hassan Arbabi, Igor Mezić. Ergodic theory, Dynamic Mode Decomposition and Computation of Spectral properties of the Koopman operator. 2017.