= Solution
Assume first that $f$ and $g$ are <integrable functions>, so that <Fubini's theorem> justifies changing the order of the two <integrals>. In the defining <Fourier transform> of the <convolution>, put $y=x-x'$:
$$
\begin{aligned}
\widetilde h(k)
&=\int_{\mathbb R}\int_{\mathbb R}e^{-ikx}f(x-x')g(x')\,dx'\,dx\\
&=\int_{\mathbb R}\int_{\mathbb R}e^{-ik(y+x')}f(y)g(x')\,dy\,dx'\\
&=\left(\int_{\mathbb R}e^{-iky}f(y)\,dy\right)\left(\int_{\mathbb R}e^{-ikx'}g(x')\,dx'\right).
\end{aligned}
$$
This proves the <convolution theorem> with the stated normalization:
$$
\boxed{\widetilde h(k)=\widetilde f(k)\widetilde g(k).}
$$
Back to article page
