= Solution
By the <Fourier inversion theorem> and <Fubini's theorem>,
$$
\begin{aligned}
h(x)
&=\frac1{2\pi}\int_{-\infty}^{\infty}
\widetilde f(k)\widetilde g(k)e^{ikx}\,dk\\
&=\int_{-\infty}^{\infty}f(y)
\left[\frac1{2\pi}\int_{-\infty}^{\infty}
\widetilde g(k)e^{ik(x-y)}\,dk\right]dy\\
&=\boxed{\int_{-\infty}^{\infty}f(y)g(x-y)\,dy}.
\end{aligned}
$$
\b[Thus multiplication of <Fourier transforms> corresponds to the <convolution> $h=f*g$, which is the <convolution theorem> in the convention used here.]
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