Stated for a first-order nonlinear ODE $dx/dt = F(x)$, $x(0) = x_0$, with $x(t) \in \mathbb{C}^n$ and $F : \mathbb{C}^n \to \mathbb{C}^n$, and for a (Banach) space $G$ of observables $\psi : \mathbb{C}^n \to \mathbb{C}$. What has to hold is a property of $G$ rather than of the dynamics, and there are two parts to it. The observable of interest must lie in $G$, because the readout is its expansion $g = d \cdot \Psi = \sum_j d_j \Psi_j$ in a basis $\Psi$ of $G$; and $G$ must be closed under the evolution — Katz, Muraleedharan and Alase write that in defining the Koopman operator they "assumed that the image of $K_t$ is in $G$, which can be ensured in practice by carefully selecting a sufficiently large $G$". Truncation is by projection onto an $N$-dimensional subspace $G_N$, and the paper is explicit that projecting does not commute with the dynamics: $\Psi^{(N)}$ and $\Pi\Psi$ agree at $t = 0$, but "in general $\Psi^{(N)}(t) \neq \Pi\Psi(t)$ for $t > 0$". Convergence of the readout is stated conditionally at this generality and not proved — "As $N \to \infty$, if $g_N(t) \to g(t)$, then the value of the observable obtained using the truncated lifted dynamics $g_N(t)[x_0] = g_N(x(t)) \to g(t)[x_0] = g(x(t))$ converges to the desired value." Every quantitative hypothesis in the paper — the dissipativity condition, the finite-time bound, the truncation-error bounds — is derived for the Fourier basis and belongs to the record below rather than here.

$F$, $y_{\mathrm{in}}$, $T$, $\varepsilon$, and a truncation or lift parameter (Carleman truncation level $N$, a phase-space grid, the level-set dimension, the homotopy order).

1. Nonlinear initial-value problem → Linear ODE systemevolve observables, project the generator

   The Koopman operator of $dx/dt = F(x)$, $x(0) = x_0$, acts on a space $G$ of observables by $(K_t\psi)(x(0)) = \psi(x(t))$ and is a linear operator on $G$; its infinitesimal generator is $L(\psi) = \lim_{t \to 0^+}(\psi(t) - \psi(0))/t$. assumption: The image of $K_t$ lies in $G$ — assumed in the definition, and "can be ensured in practice by carefully selecting a sufficiently large $G$" — and the observable of interest lies in $G$ as well, since the readout is its expansion $g = d \cdot \Psi = \sum_j d_j \Psi_j$ in the basis. Writing $L\Psi_j = \sum_k L_{kj}\Psi_k$ in a basis $\Psi$ of $G$ represents $L$ as $[L_{jk}]$ and gives the Koopman representation $d\Psi(t)/dt = L^{T}\Psi$ — transposed because what evolves is the basis of $G$ rather than the representation of one vector in it. Evaluated at an initial configuration it reads $d(\Psi(t)[x_0])/dt = L^{T}(\Psi(t)[x_0])$, so the lifted state is an array of complex numbers, and the target comes back out as $g(x(t)) = d \cdot \Psi(t)[x_0]$. approximation: $G$, and with it $L$, is in general infinite-dimensional, so the generator is replaced by its projection onto an $N$-dimensional subspace $G_N$ spanned by $\{\Psi_j\}_{j=1}^{N}$: $L_N$ has $j$-th column $\Pi L\Psi_j$ in that basis, and the lifted dynamics becomes $d\Psi^{(N)}/dt = L_N^{T}\Psi^{(N)}$. That is not the projected exact equation $\Pi L\Psi_j = \Pi\, d\Psi_j/dt$, so although $\Psi^{(N)}$ and $\Pi\Psi$ coincide at $t = 0$, in general $\Psi^{(N)}(t) \neq \Pi\Psi(t)$ for $t > 0$, and the observable is read from the closest approximation in $G_N$, $g \approx g_N = \sum_{j=1}^{N} d_j\Psi_j$.

   approximationassumption

A linear generator with any inhomogeneity, a lift map, a readout map, and an error bound as a function of the truncation parameter.

given  a first-order nonlinear ODE  dx/dt = F(x),  x(0) = x_0,  x(t) in C^n,
       an observable g : C^n -> C wanted at the final time,
       a truncation level N

# the parameter of this method is the pair (G, Psi), not a step size
choose  a space of observables G containing the quantity of interest g
choose  a basis Psi = {Psi_j} of G
        # G fixes which observables the lifted dynamics can report
        # Psi fixes the structure of the generator
        # so this is a family of lifts parameterised by that choice,
        # not a single lift

# the Koopman generator, written in that basis
        # (K_t psi)(x(0)) = psi(x(t)),  and K_t is linear on G
        # L(psi) = lim_{t -> 0+} (psi(t) - psi(0)) / t
form    L = [L_jk]   from   L Psi_j = sum_k L_kj Psi_k
write   dPsi(t)/dt = L^T Psi
        # transposed: what evolves is the basis of G, not one vector's
        # representation in it
evaluate at the initial configuration:
        d(Psi(t)[x_0])/dt = L^T (Psi(t)[x_0])
        # the lifted state is now an array of complex numbers

# truncate by projecting onto N basis functions
project onto  G_N = span{Psi_1, ..., Psi_N}  by  Pi : G -> G_N
form    L_N,  whose j-th column is  Pi L Psi_j  in the basis {Psi_j}_{j=1..N}
        # if Pi keeps Psi_j for j <= N and sends it to 0 for j > N,
        # L_N^T is simply the top-left N x N block of L
solve   dPsi^(N)/dt = L_N^T Psi^(N),   initial data Psi^(N)(x_0)
        # not the projected exact equation: Psi^(N)(t) != Pi Psi(t) in
        # general for t > 0, though the two agree at t = 0

# decode
expand  g ~ g_N = sum_{j=1..N} d_j Psi_j      # the closest approximation in G_N
        # obtaining Psi(t)[x_0] at time t also gives g(x(t)) = d . Psi(t)[x_0]

hand the truncated linear ODE, its initial data Psi^(N)(x_0) and the decoding d
     to the layer below