How to detect interdependence of functions Let $f,g:\mathbb{R}\to\mathbb{R}$ be two functions (assume them to be regular enough, let's say $L^2(\mathbb{R})$). It might be the case that these functions are not totally independent. For example we could have that the value of $f(x)$ depends in some way on the value of $g(x-t)$. How is it possible to detect such interdependences?

A first idea I had was to look at the Fourier transform of the two functions. Then if $f(x)$ depends somewhat strongly (e.g. linearly) on $g(x)$, the transforms should be basically the same except for multiplication by some factor (by the usual Fourier transform of a shifted function rule).

Another thing I found around is cross-correlation, but this does not work well for non-linear phenomena.

Is my first idea viable? And are there good references on the subject around?

Also, what are the correct tags for this question?
FunctionalDependence

page 17

Criterion for functional independence of $ u(x,y), v(x,y) $ is through the Jacobian

$$ J= \dfrac{\partial (u(x,y),v(x,y))}{\partial (x,y)} \ne 0 $$

In the event of vanishing J, function defining transformation $u$ to $v$ can be also found out.
FunctionalDependence

page 17

Criterion for functional independence of $ u(x,y), v(x,y) $ is through the Jacobian

$$ J= \dfrac{\partial (u(x,y),v(x,y))}{\partial (x,y)} \ne 0 $$

In the event of vanishing J, function defining transformation $u$ to $v$ can be also found out.