Say I have discrete-time nonlinear dynamics:
$$ \mathbf{x}_{t+1}=f(\mathbf{x}_t) $$ where $\mathbf{x}\in\mathbb{R}^d$.
I have an observable function, as an example, defined as:
$$ g(\mathbf{x}) = \left[ \cos(\mathbf{w}_0\cdot\mathbf{x}), \ldots, \cos(\mathbf{w}_i\cdot\mathbf{x}),\ldots, \cos(\mathbf{w}_n\cdot\mathbf{x})\right]^\top $$ $$n\rightarrow \infty$$ where $\mathbf{w}_i \sim \mathcal{N}(0,\sigma^2\mathbf{I}) $. Basically, $g$ maps $\mathbf{x}$ to an infinite dimensional feature vector.
Does the Koopman theory imply that there exists a linear operator $\mathbf{K}$ such that
$$ g(\mathbf{x}_{t+1}) = \mathbf{K} g(\mathbf{x}_t) $$ ?
or does the linear dynamics exist only if I use a $g$ that is specifically constructed for the given nonlinear dynamics $f$, instead of an arbitrary $g$ that I show above as an example? In other words, I am questioning would an arbitrary nonlinear map $g$ make the dynamics linear in the feature space regardless of the given $f$.
