= Solution
First take $f,g\in L^1(\mathbb R)$, so <Fubini's theorem> applies to the <cross-correlation> double integral. With $s=x+y$, the <Fourier transform> is
$$
\begin{aligned}
\widehat h(k)&=\int_{\mathbb R}\int_{\mathbb R}e^{-ikx}\overline{f(y)}g(x+y)\,dy\,dx\\
&=\left[\int_{\mathbb R}\overline{f(y)}e^{iky}\,dy\right]\left[\int_{\mathbb R}g(s)e^{-iks}\,ds\right]\\
&=\boxed{\overline{\widehat f(k)}\,\widehat g(k)}.
\end{aligned}
$$
The <complex conjugation> and the sign in $x+y$ are both essential to this convention.
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