Suppose you have one real-valued signal $s(t)$ and another the real-valued signal $r(t)$ related to $s(t)$ by a linear integral transformation with a real kernel $k$ such that $r(t) = \int dt'\, k(t-t') s(t')$. The kernel depends on particulars of the problem, e.g. boundary conditions or the wave equation governing the physical system we're modeling.
Instead of working with real-valued signals, we can work with complex-valued signals, and then at the end of the day take the real part.
The analytic signals $s_a(t)$ and $r_a(t)$ related to the real-valued signals are given by $x_a(t)=x(t)+i\cal H[x](t)$ where $\cal H$ is the Hilbert transform and $x$ stands for $s$ or $r$.
Given $s(t)$, we can calculate $r(t)$ in two ways:
We could use the integral transformation with kernel $k$.
We could calculate $s_a(t)$ from $s(t)$, then perform an (so far unknown) integral transformation $r_a(t) = \int dt'\, h(t-t') s_a(t')$ with kernel $h$, and then take the real part of $r_a(t)$.
Obviously, the results must be the same, but what must $h$ be? In particular, how is $h$ related to $k$?
