The Fourier transform of the product of two functions f(x) and g(x) is given as:

$\mathcal{F}[ f(x)g(x)] = \int_{-\infty}^{+\infty} F(\omega^\prime) G(\omega - \omega^\prime) d\omega^\prime \; = \; \mbox{convolution of} \; \; F(\omega^\prime )G(\omega^\prime)$

where $F(\omega^\prime)$ and $G(\omega^\prime)$ are the Fourier transforms of $f(x)$ and $g(x)$ respectively.

Although I understand the derivation of this formula, I've got difficulty making sense of two frequency terms $\omega$ and $\omega^\prime$. I'm fine with $\omega^\prime$ but I don't know what to make of $\omega$. Should I treat it as a constant, or should I set it to zero?

I'm really interested in the Fourier transform of the square of the second derivative of a function e.g. $\mathcal{F}[ f^{\prime\prime}(x)^2 ]$. Because this problem does not involve a shift, I don't know what to make of the shift term $\omega$.