Making sense of the Fourier transform of the product of two functions

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

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

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. $mathcal{F}[ f^{primeprime}(x)^2 ]$. Because this problem does not involve a shift, I don't know what to make of the shift term $omega$.

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