Note that when $f$ is bijective from $(-R,R)$ to $(-\pi, \pi)$, the above representation formula does not require the use of the cut-off function $\chi$ and allows describing all solutions $y$ of for all initial data in $y^0 \in \mathfrak{C}(R/\tau)$, see Remark \[Remark-a\] for further details.

We now give some informal arguments motivating the representation formula in Theorem \[Thm-DiscTransmutation\].

First, we remark that under the assumptions on the operator ${{\mathcal{A}}}$, solutions $y$ of admit the following spectral decomposition: $$\label{Y-expanded}
    y(t) = \sum_j a_j e^{i \mu_j t} \Phi_j,$$ where $(a_j)$ are the coefficients of the initial datum on the basis $\Phi_j$: $$\label{Init-Data}
    y^0 = \sum_j a_j \Phi_j.$$ Similarly, due to , the solutions $y_\tau$ of the time-discrete system with initial datum $y^0$ as in can be written as $$\label{Y-tau-Expanded}
    y^k_\tau = \sum_j a_j e^{i \mu_{j,\tau} k \tau} \Phi_j,$$ where the $\mu_{j,\tau}$ are defined by . Obviously, we can then extrapolate a time-continuous version $z_\tau$ of $y_\tau$: $$\label{Z-cont}
    z_\tau(s) = \sum_j a_j e^{i \mu_{j,\tau} s} \Phi_j.$$ Here, we use the notation $s$ for the continuous time-variable corresponding to the *a priori* time-discrete dynamics, and $z_\tau$ for the new state variable to avoid confusion with time $t$ and state $y$ of the continuous 