any function $v \in L^2(-\pi/\tau, \pi/\tau)$, we define the inverse Fourier transform ${{\mathcal{F}}}_\tau^{-1}[v]$ at scale $\tau>0$ as: $$\label{FourierDisInv}
                {{\mathcal{F}}}_\tau^{-1}[v](k \tau) = \frac{1}{2\pi} \int_{-\pi/\tau}^{\pi/\tau} v(\mu_\tau) \exp(i \mu_\tau k \tau) \ d\mu_\tau, \quad k \in \mathbb{Z}.$$

According to Definition \[DiscreteFourier\], one easily checks that these transforms are inverse one from another, so that, $$\label{FourierIso}
        \left\{ \begin{array}{l}
            {\displaystyle}{{\mathcal{F}}}_\tau^{-1}[{{\mathcal{F}}}_\tau[u_\tau]] (k \tau ) = u_\tau (k \tau), \quad k \in {\mathbb{Z}},
            \smallskip
            \\
            {\displaystyle}{{\mathcal{F}}}_\tau[{{\mathcal{F}}}_\tau^{-1}[v]] (\mu_\tau) = v(\mu_\tau),\quad  \mu_\tau \in (-\pi/\tau,\pi/\tau), 
        \end{array}\right.$$

Similarly as for the continuous Fourier transform, we also have the following Parseval identity: $$\label{ParsevalDis}
            \frac{1}{2\pi} \int_{-\pi/\tau}^{\pi/\tau} |{{\mathcal{F}}}_\tau [u_\tau](\mu_\tau)|^2 \ d \mu_\tau = \tau \sum_{k\in\mathbb{Z}} |u_\tau(k\tau)|^2.$$ These properties will be used in the sequel.

In the following, for a Hilbert space $H$, the space $L^2(\tau {\mathbb{Z}}; H)$ is the set of discrete functions $u_\tau$ defined on $\tau {\mathbb{Z}}$ with values in $H$ endowed with the norm $${\left\Vert u_\tau\right\Vert}_{L^2(\tau {\mathbb{Z}}; H)}^2 = \tau \sum_{k \in {\mathbb{Z}}} {\left\Vertu_\tau(k \tau)\right\Vert}_H^2.$$

Proof of Theorem \[Thm-DiscTransmutation\]
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Expand $y^0$ as $$y^0 = \sum_{j, \ |\mu_j| \leq \delta/\tau} a_j \Phi_j.$$ Then, for all $k \in {\mathbb{Z}}$, according to , $$y_\tau^k = \sum_{j, \ |\mu_j| \leq \delta/\tau} a_j \Phi_j \exp (i \mu_{j,\tau} k\tau),$$ and thus $y(t)$ defined by can be written 