Suppose I have a function of the form:
$G(t,x) = \alpha\left(P(t,x) - \Theta(t,x) \right)^+ + \beta \left( P(t,x) - \Theta(t,x) \right)^-$.
Here, $P(t,x)$ and $\Theta(t,x)$ have compact support and are functions that "behave nicely". Also, $x^+ = max(x,0)$ and $x^- = min(x,0)$ and $\alpha,\beta \in \mathbb R$ are constants.
I'd like to know if the fourier transform of G(t,x) (in the x variable) can be written in terms of the fourier transforms of $\Theta$ and $P$.
i.e. If $\mathcal F \left[ G(t,x) \right] = \hat G(t,w)$, $\mathcal F \left[ \Theta(t,x) \right] = \hat \Theta(t,w)$ and $\mathcal F \left[ P(t,x) \right] = \hat P(t,w)$, can I say that $\hat G(t,w) = f(t,\hat G(t,w), \hat \Theta(t,w))$?
Would I need any further restrictions on any of the functions I'm using for this to be possible?
Let me know if any clarifications are needed.