Convolution theorem

For $\hat{f}(\xi )=\int f(x)e^{-2\pi ix\cdot \xi }dx$, Fubini gives $\widehat{f\ast g}=\hat{f}\hat{g}$.

For $F(x)=e^{-x}{1}_{x\ge 0}$, induction in the convolution integral givesIts Fourier transform in the angular-frequency convention is $(1+ik{)}^{-n}$.