We are given $x_1,x_2 \in \mathbb{R}$ and we want to find two functions $v_1(t),v_2(t)$ such that:
$x_1x_2 = \int_{-\infty}^{\infty} v_1(t)-v_2(t) dt$ A very interesting restriction that we have is that the object generating $v_1(t)$ only knows $x_1$, while $v_2(t)$ is generated only by knowing $x_2$.
The application of this is like this. We have two ends of a wire with some component Y in between. We call one end as $1$ where $x_1$ is known and second end as $2$ where $x_2$ is known. We want to send a signal from both ends which gets aggregated at Y, but we want to choose two signals $v_1(t),v_2(t)$ such that when Y sums them up the sum of the two signals becomes equal to the multiplication of $x_1$ and $x_2$.