= Fourier transform of a tempered distribution
{c}
{title2=$\widehat u$}
With $\widehat\varphi(\xi)=\int e^{-ix\cdot\xi}\varphi(x)\,dx$, the Fourier transform of a <tempered distribution> is defined by
$$
\langle\widehat u,\varphi\rangle=\langle u,\widehat\varphi\rangle.
$$
Continuity of the <Fourier transform> on the <Schwartz space> makes this well defined. For regular integrable functions, <Fubini's theorem> shows that it agrees with the ordinary transform. The inverse convention has factor $(2\pi)^{-n}$ and the positive exponential.
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