= Fourier product formula
{c}
{title2=$\widehat{fg}=(2\pi)^{-1}\widehat f*\widehat g$}
= Fourier transform of a pointwise product
{c}
{synonym}
For <Schwartz functions> $f,g$ and the <Fourier transform> convention $\widehat f(\xi)=\int f(x)e^{-ix\xi}\,dx$, the displayed formula exchanges a pointwise product for a <convolution>. Substitute <Fourier inversion> for $g$ into the <Fourier transform> of $fg$. <Fubini's theorem>, justified by $\|f\|_1\|\widehat g\|_1<\infty$, gives
$$
\widehat{fg}(\xi)=\frac1{2\pi}\int\widehat g(\eta)\widehat f(\xi-\eta)\,d\eta.
$$
Unlike the <convolution theorem> $\widehat{f*g}=\widehat f\widehat g$, this version cannot be asserted for arbitrary pairs in $L^1$ without additional hypotheses, since their pointwise product need not be an <integrable function>.
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