= Solution
For fixed real $t$, <multiplication> by $e^{itx}$ does not change the absolute value. The product integrability proved in (ii) therefore justifies <Fubini theorem> in the following calculation. <Set> $u=x-y$ in the inner <integral>:
$$
\begin{aligned}
\widehat{g*h}(t)
&=\int_{\mathbb R}\int_{\mathbb R}g(x-y)h(y)e^{itx}\,dy\,dx\\
&=\int_{\mathbb R}h(y)\left[\int_{\mathbb R}g(x-y)e^{itx}\,dx\right]dy\\
&=\int_{\mathbb R}h(y)e^{ity}\left[\int_{\mathbb R}g(u)e^{itu}\,du\right]dy\\
&=\widehat g(t)\widehat h(t).
\end{aligned}
$$
Thus
$$
\boxed{\widehat{g*h}=\widehat g\,\widehat h.}
$$
The positive exponential convention in the source is used throughout; no normalization factor is introduced under that convention.
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