$\delta < R$, $$\label{HypG-3}
     \inf\{f'(\alpha); \ |\alpha| \leq \delta \} >0$$ to ensure the invertibility of $f$.

We finally define $g : (-f(R), f(R)) \to {\mathbb{R}}$ as the inverse function of $f$ on $(-R, R)$. This can be done due to assumption .

Before going further, let us point out that several classical time-discretization schemes fit the abstract setting provided by Assumptions ––, see Section \[ExamplesTimeDisc\] for several examples.

A representation formula
------------------------

We are now in position to state the following result, whose proof can be found in Section \[Sec-Rho-tau\]:

\[Thm-DiscTransmutation\] Let $f$ be a smooth function describing the time discrete operator ${\mathbb{T}}_\tau$ as in , assume ––, and fix $\delta \in (0,R)$.

Let $\chi$ be a $C^\infty$ function compactly supported in $(-f(R), f(R))$ and equal to $1$ in $(-f(\delta), f(\delta))$. For $\tau >0$, define then $\rho_\tau(t,s)$ for $t \in {\mathbb{R}},\ s \in {\mathbb{R}}$ as $$\label{KernelDiscTransmutation}
        \rho_\tau(t, s) = \frac{1}{2\pi} \int_{- \pi/\tau}^{\pi/\tau} \exp\left(-\frac{i g(\mu_\tau \tau) t}{\tau} \right) \chi\left(\mu_\tau\tau\right) e^{i \mu_\tau s}\, d\mu_\tau.$$ Then, if $y^0 \in \mathfrak{C}(\delta/\tau)$ and $y^k_\tau$ is the corresponding solution of , the function $y(t)$ defined by $$\label{Transmutation}
        y(t) = \tau \sum_{k \in {\mathbb{Z}}} \rho_\tau(t, k \tau) y_\tau^k$$ is the solution of with initial data $y^0$.

