geometric settings still enjoy observability properties, see e.g. [@BurqZworski].

Outline
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The article is organized as follows. In Section \[ExamplesTimeDisc\], we give several instances of classical time-discretization schemes that fit the abstract setting with a function $f$ satisfying Assumptions ––. Section \[Sec-Rho-tau\] is devoted to the proofs of Theorem \[Thm-DiscTransmutation\] and Theorem \[Thm-Main\]. In Section \[Sec-Reverse\] we present a reverse representation formula and discuss its application to uniform admissibility results and use it to prove the optimality of the time estimate . In Section \[Sec-Further\], we present some further comments. In Section \[Sec-Open\], we end up discussing some open problems.

Some admissible time-discretization schemes {#ExamplesTimeDisc}
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In this section, we provide several classical time-discretization schemes that fit the setting – and satisfy assumptions ––.

The midpoint scheme
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Perhaps the simplest time-discretization of which preserves the energy is the midpoint scheme: for $\tau >0$, the time-discrete equation is given by: $$\label{Midpoint}
        \frac{y_\tau^{k+1} - y_\tau^k}{\tau} = {{\mathcal{A}}}\left(\frac{y_\tau^k + y_\tau^{k+1}}{2} \right), \quad k \in {\mathbb{Z}}, \qquad y_\tau^0 = y^0.$$ Thus, if $\Phi_j$ is an eigenvector of ${{\mathcal{A}}}$ with eigenvalue $i \mu_j$, the solution $y_\tau$ of with initial data $y^0 = \Phi_j$ is given by $$\begin{gathered}
        y_\tau^k = \left(\frac{1 + i \mu_j \tau 