= Solution
A <tempered distribution> is a continuous <linear functional> $u:\mathcal S(\mathbb R)\to\mathbb C$. Equivalently, for some integer $N$ and constant $C$,
$$
|\langle u,\phi\rangle|\leq C\sum_{0\leq m,n\leq N}p_{m,n}(\phi)
\qquad(\phi\in\mathcal S(\mathbb R)).
$$
The <Fourier transform of a tempered distribution> is defined by
$$
\boxed{\langle\widehat u,\phi\rangle:=\langle u,\widehat\phi\rangle.}
$$
The <Fourier bounds for Schwartz functions> make the right side a continuous <linear functional> on the <Schwartz space>, so $\widehat u$ is again a <tempered distribution>.
For an <integrable function> $f$, the <regular distribution> $u_f$ acts by $\langle u_f,\phi\rangle=\int f\phi$. The estimate $|\langle u_f,\phi\rangle|\leq\|f\|_1p_{0,0}(\phi)$ proves it is a <tempered distribution>. Since $\iint|f(x)\phi(\xi)|\,dx\,d\xi<\infty$, <Fubini's theorem> gives
$$
\langle\widehat{u_f},\phi\rangle
=\int f(x)\int\phi(\xi)e^{-ix\xi}\,d\xi\,dx
=\int\widehat f(\xi)\phi(\xi)\,d\xi
=\langle u_{\widehat f},\phi\rangle.
$$
Thus the <distributional Fourier transform> agrees with the ordinary <Fourier transform>.
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