that $f$ is $C^\infty$ and satisfies $$\label{HypG-1}
    f(0) = 0, \quad f'(0) = 1.$$ This assumption on $f$ is satisfied for most of the time-discretization approximation schemes: Roughly speaking, this is equivalent to the consistency of the time-discretization.

To simplify notations and avoid technical developments, we furthermore assume that $f$ is odd. This is not necessary in our arguments, but in practice, most of the time-discretization schemes modeled by fit in this class.

We also assume that $f$ is real valued and $$\label{HypG-2}
    f : (-R, R) \to (-\pi, \pi),$$ where $R \in {\mathbb{R}}_+^* \cup \{\infty\}$ may be infinite. The fact that $f$ is real-valued is equivalent to the fact that the norms of solutions of are constant in the discrete time $k \in {\mathbb{Z}}$ and ensures the stability of the numerical scheme. The range of $f$ is limited to $(-\pi, \pi)$ to avoid aliasing since one cannot measure oscillations at frequencies higher than $\pi/\tau$ on a discrete mesh of mesh-size $\tau$.

In particular, for to be well-defined, we always consider solutions of with initial data lying in some class of filtered data at the scale $\tau$, namely $\mathfrak{C}(\delta/\tau)$ for some $\delta \in (0,R)$.

We also assume that for all 