dynamics.

Transmutation or subordination refers to the possibility of expressing one semi-group as a function of another one. In our context, this simply consists in writing the solutions $y$ of as a function of $z_\tau(s)$ by means of a kernel $\tilde \rho = \tilde \rho(t,s)$ under the form $$\label{GeneralTransForm}
    y(t) = \int_{\mathbb{R}}\tilde \rho_{\tau} (t,s) z_\tau(s) \, ds, \quad t \in {\mathbb{R}}.$$ For this to be done, using the explicit expressions and , the kernel $\tilde \rho(t,s)$ has to be built so that $$\label{Formal-tilde-rho-req}
    e^{i \mu_j t} = \int_{\mathbb{R}}\tilde \rho_\tau(t,s) e^{i \mu_{j,\tau} s} \, ds, \quad t \in {\mathbb{R}}, \, \mu \in {\mathbb{R}}.$$ Interpreting the right-hand side of as a Fourier transform in $s$ and taking into account that $\mu_{j,\tau} = f (\mu_j \tau)/\tau$ with $f$ invertible, naturally leads to $$\label{Formal-tilde-rho}
    \tilde \rho_\tau(t,s) = \frac{1}{2 \pi} \int_{\mathbb{R}}\exp\left(-\frac{i g(\mu \tau) t}{\tau} \right) e^{i \mu s}\, d\mu.$$ This formula is a simplified version of the one of $\rho_\tau$ in , in which the above time-continuous function $z_\tau$ in has to be replaced by the time-discrete function $y_\tau$ in and the filtering operator $\chi$ has been introduced.

Application to observability
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As an application of Theorem \[Thm-DiscTransmutation\], we consider an observation problem for the 