I have an expression:
$c(s) = \int_{-\infty}^{\infty} d\omega e^{-i*\omega*s}*F(\omega)$
$G(W)=\int_{0}^{\infty} ds e^{i*W*s}*c(s) $
Is $G(W) = F(W)$? What is the relation of G(W) and F(W)?
$G(W)$ looks like a backward Fourier transform of a forward Fourier transform so it should be related to $F(W)$. The only problem here is that the limit of the back Fourier transform is not from $-\infty$ to $\infty$, it's from zero to infinity.