I am a physicist working on a micromechanics problem and have run into the following mathematical issue:
I need to find the Fourier transform of the derivative of the squared convolution, i.e.: $$ \mathcal{F} \left\{ \frac{\partial}{\partial x} \left[f(x) *g(x)\right]^2 \right\} $$
My simple approach was to expand the square and use the chain rule:
$$ \mathcal{F} \left\{ \frac{\partial}{\partial x} \left[f(x)*g(x) \cdot f(x)*g(x)\right] \right\} = \mathcal{F} \left\{\frac{\partial}{\partial x}[f(x)*g(x)] \cdot f(x)*g(x) + f(x)*g(x) \cdot \frac{\partial}{\partial x}[f(x)*g(x)]\right\}$$
Then we can move the derivative into the convolution: $$2 \mathcal{F} \left\{[f'(x)*g(x)] \cdot [f(x)*g(x)]\right\}$$
The partial derivative is now effectively out of the picture, but this where I get stuck - is there a way to eliminate or reduce the number of Fourier transforms I need to do in order to solve (read: compute) this?
Thanks for your help!
Cheers, Hugh