$y=G(x)$ is translation invariant, $G\Bigl(T_sx\Bigr)(t)=(T_sy)(t)=y(t+s)$ Together with the linearity this has the consequence that also differential operators are preserved,
$$\dot y(t)=\lim_{s\to 0}\frac{(T_sy)(t)-y(t)}s=\lim_{s\to 0}\frac{G\bigl(T_sx\bigr)(t)-G\bigl(x\bigr)(t)}{s}=G\left(\lim_{s\to 0}\frac{T_sx-x}s\right)(t)=G\bigl(\dot x\bigr)(t).$$
Now you can also apply this to the oscillator equation, $G(\ddot x+\omega^2x)=\ddot y+ω^2y$ and if $x$ is sinusoid with frequency $ω$, then so is $y$.
With $$G(\cos(ω\,\cdot\,))(t)=a\cos(ωt)+b\sin(ωt)$$ you also get the shifted
$$
G(\sin(ω\,\cdot\,))=G(\cos(ω\,\cdot\,-\tfrac\pi2))(t)=a\sin(ωt)-b\cos(ωt)
$$
so that indeed there are only two free parameters per frequency. To get the attenuation and phase, you only need to compute the polar coordinates $(A,\varphi)$ of the point $(a,-b)$.
To answer these comments in order:
Think of a network of nodes (like a neural net or a Hopfield net) which are connected by symmetrical bidirectional connections. If two nodes that have an activation of 1 are connected by a weight of 1, then they reinforce each other. If they are connected by a weight of -1, then they inhibit each other. If two nodes have an activation of -1, then again, a weight of 1 means reinforcement. If one node has an activation of 1, and the other has an activation of -1, then a weight of 1 between them would be inhibitory.
People set up these networks, and let them evolve over time, and try to get the most activations that are consistent with each other. The network doesn't always settle into the best solution, but it can settle into a almost-good solution.
You can express networks like these with equations such as the above (A + B - AB + CD - CA.....
So you need to be able to simulate both sums and products, including products with negative signs. My idea is that somewhere in the interference pattern there is a maximal solution (maybe it would appear darkest on a photo, for instance). The nice thing about an interference pattern is that it tries out all combinations (I think). But can it be done?
An equation such as A + B - C can be expressed as waves - A and B can be the source of waves that are in phase and with the same frequency, and C can be the source of a wave that is 90 degrees out of phase with the others. But how do you express the products (AB, CD, etc)?
On the last comment, are you saying that you generate any arbitrary wave shape electronically?
Best Answer
If you mean that you want a periodic function whose period looks like a "distorted" sine, you can do that by altering the argument to sine. Normally we look at $\sin (t)$, where $t$ increases steadily. But if we alter this slightly, we can take $\sin (s(t))$, where $s(t)$ has a graph that looks like the graph of $t \mapsto t$, but slightly distorted. A good example: $$ t \mapsto \sin ( t + 0.2 \sin t) $$ Graph:
Same idae, but with $0.2$ replaced by $0.5$: