Can it be proven that such functions don't exist?

Question

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 $x_2$.

Answer 1

It cannot be done. You are looking for two functions $u:\ (x,t)\mapsto u(x,t)$ and $v:\ (y,t)\mapsto v(y,t)$ such that $$x \cdot y\ \equiv \ \int_{-\infty}^\infty\bigl(u(x,t)-v(y,t)\bigr)\ dt$$ for all $(x,y)$ in some domain $\Omega\subset{\mathbb R}^2$. It follows that for any two $x_1\ne x_2$ and any $y$ we should have $$(x_1-x_2)y\ =\ \int_{-\infty}^\infty \bigl(u(x_1,t)-u(x_2,t)\bigr)\ dt\ .$$ This is impossible, as the RHS is constant with respect to $y$.

Answer 2

I've deleted my original answer, since Christian Blatter's nice and succint proof is better and works even without any assumptions about the existence and finiteness of the individual integrals $\int_{-\infty}^\infty v_1(t) \;dt$ and $\int_{-\infty}^\infty v_2(t) \;dt$. However, I've left in the following addendum:

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What you can do, however, is have

$$x_1 x_2 = \exp \left( \int_{-\infty}^\infty v_1(t) \;dt - \int_{-\infty}^\infty v_2(t) \;dt \right) = \exp \left( \int_{-\infty}^\infty v_1(t) - v_2(t) \;dt \right),$$

with $v_1$ and $v_2$ chosen arbitrarily such that $\int_{-\infty}^\infty v_1(t) \;dt = \log x_1$ and $\int_{-\infty}^\infty v_2(t) \;dt = -\log x_2$. Of course, this particular solution only works for positive $x_1$ and $x_2$. Choosing appropriate functions $v_1$ and $v_2$ is left as an exercise, although obviously e.g. pulses with unit width and amplitudes $\log x_1$ and $-\log x_2$ will do.