\tau}^{1/2}\left(\frac{\varphi_\tau^0+\varphi_\tau^{1}}{2}\right)  \right\Vert}_{{{\mathcal{X}}}_0}^2 
        \\
        \leq C \tau \sum_{ k \tau \in (0,T)} {\left\Vert{{\mathcal{B}}}_1 {{\mathcal{A}}}_0^{-1/2} {{\mathcal{A}}}_{0, \tau}^{1/2}\left(\frac{\varphi_\tau^k + \varphi_\tau^{k+1}}{2}\right)\right\Vert}_{{\mathcal{U}}}^2.
    \end{gathered}$$ Now, applying this identity to ${{\mathcal{A}}}_0^{1/2}{{\mathcal{A}}}_{0, \tau}^{-1/2} \varphi_\tau$, which is still a solution of with initial data in $\mathfrak{C}(\delta/\tau)$, $$\begin{gathered}
            {\left\Vert{{\mathcal{A}}}_0^{1/2} {{\mathcal{A}}}_{0, \tau}^{-1/2}\left( \frac{\varphi_\tau^{1}-\varphi_\tau^0}{\tau}\right)\right\Vert}_{{{\mathcal{X}}}_0}^2 
            + {\left\Vert {{\mathcal{A}}}_0^{1/2} \left(\frac{\varphi_\tau^0+\varphi_\tau^{1}}{2}\right)  \right\Vert}_{{{\mathcal{X}}}_0}^2 
            \\
            \leq C \tau \sum_{ k \tau \in (0,T)} {\left\Vert{{\mathcal{B}}}_1 \left(\frac{\varphi_\tau^k + \varphi_\tau^{k+1}}{2}\right)\right\Vert}_{{\mathcal{U}}}^2.
    \end{gathered}$$ But easy spectral computations show that, in the class $\mathfrak{C}(\delta/\tau)$, there exists a constant $C>0$ depending only on $\delta$ such that $$\tilde E^{1/2} \leq C   \left({\left\Vert{{\mathcal{A}}}_0^{1/2}{{\mathcal{A}}}_{0, \tau}^{-1/2}\left( \frac{\varphi_\tau^{1}-\varphi_\tau^0}{\tau}\right)\right\Vert}_{{{\mathcal{X}}}_0}^2 
            + {\left\Vert{{\mathcal{A}}}_0^{1/2} \left(\frac{\varphi_\tau^0+\varphi_\tau^{1}}{2}\right)  \right\Vert}_{{{\mathcal{X}}}_0}^2\right).$$

A representation formula, properties of the kernel $\rho_\tau$ and applications {#Sec-Rho-tau}
===============================================================================

We will first recall basic facts on the discrete Fourier transform. We will then prove Theorem \[Thm-DiscTransmutation\] and give some estimates on the kernel $\rho_\tau$ in , which we use in Section \[Sec-Proof-Main\] to prove Theorem \[Thm-Main\].

Discrete Fourier transforms
---------------------------

Let us introduce the definition of the discrete Fourier and inverse Fourier transforms:

\[DiscreteFourier\] Given any function $u_\tau$ defined on $\tau {\mathbb{Z}}$, we define its discrete Fourier transform ${{\mathcal{F}}}_\tau[u_\tau]$ at scale $\tau$ as: $$\label{FourierDis}
            {{\mathcal{F}}}_\tau [ u_\tau](\mu_\tau) = \tau \sum_{k \in \mathbb{Z}} u_\tau(k \tau) \exp(- i \mu_\tau k \tau), \quad \mu_\tau \in (-\pi/\tau,\pi/\tau).$$ For 