I have two functions $f(x)$ and $g(x)$ where $f(x)=g(x)=0$ for $x<0$. I want to use the convolution theorem to get $\tilde{f}(\omega)\tilde{g}(\omega)$, and for some reason I only want the imaginary part of $\tilde{g}(\omega)$, i.e. $\tilde{f}(\omega)iIm\big(\tilde{g}(\omega)\big)$. To do that I use the fact that $FT(g(x)-g(-x))=2iIm\big(\tilde{g}(\omega)\big)$, which means that the convolution is now between $f(x)$ and $\frac{1}{2}(g(x) - g(-x))$. But since $g(-x)=0$ for $x>0$ now and $f(x)=0$ for $x>0$, this means that we can simply discard $g(-x)$ in the convolution. So we are back to square one.
What is the reason behind the inconsistency?