= Solution
Choose the <Fourier transform> convention
$$
\boxed{\widehat f(\xi)=\int_{\mathbb R^n}e^{-ix\cdot\xi}f(x)\,dx,\qquad
f(x)=(2\pi)^{-n}\int e^{ix\cdot\xi}\widehat f(\xi)\,d\xi.}
$$
It maps the <Schwartz space> to itself. For a <tempered distribution> $u$, define the transform by bilinear distributional duality:
$$
\boxed{\langle\widehat u,\phi\rangle=\langle u,\widehat\phi\rangle\qquad(\phi\in\mathcal S).}
$$
For an ordinary integrable function this agrees with the first formula by interchanging its integrals. The absence of complex conjugation is the distribution-pairing convention.
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