= \delta_0(s)$ is simply the Dirac delta transported at velocity one: $\rho_*(t,s) = \delta_{t} (s)$. Of course, for the kernel under consideration, $\partial_s^\tau$ is not a classical differential operator, but it is nevertheless very close to the classical derivation operator $\partial_s$ at low frequency since $g'(0) = 1$. We can thus interpret as a transport equation with some added dispersion effects at frequencies of the order of $1/\tau$.

Let us also point out that the representation formula in given through the kernel $\rho_\tau$ in is not far from a Fourier integral operator - see e.g. [@Treves-Book-80] - with the phase $\varphi(t,s, \mu_\tau) = \mu_\tau s- g(\mu_\tau) t$, and similar techniques can be employed. In particular the above localization result can also be seen as a counterpart of the fact that the kernel of a Fourier integral operator with phase $\varphi$ is regularizing outside of the set in which $\partial_{\mu_\tau} \varphi = 0$.

Next, in Theorem \[DiscTransmutationReverse\], we discuss a reverse representation formula giving solutions of the time-discrete equation in terms of solutions of the continuous equation . This allows us to prove admissibility results for solutions of uniformly with respect to the time-discretization parameter $\tau$, see Theorem \[MainFully5\], and the 