= Fourier transform of a periodic distribution
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{title2=$\widehat u=2\pi\sum_{n\in\mathbb Z}c_n\delta_n$}
= Distributional Fourier series of a periodic distribution
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{synonym}
For a $2\pi$ <periodic distribution> $u\in\mathcal S'(\mathbb R)$, choose a <smooth periodic partition of unity> $\Psi$, and define $c_n=(2\pi)^{-1}\langle u,\Psi e^{-inx}\rangle$. A finite <Schwartz seminorm> bound on the fixed <compact support> gives $|c_n|\leq C(1+|n|)^N$. For a <Schwartz function> $\phi$, the <Poisson summation formula> and <Fourier inversion> imply $P\widehat\phi(x)=\sum_n\phi(-n)e^{inx}$. Localizing the pairing for $u$ gives
$$
\langle\widehat u,\phi\rangle=\langle u,\Psi P\widehat\phi\rangle=2\pi\sum_n c_n\phi(n),
$$
proving the displayed <Fourier transform> formula. The series converges in <tempered distributions> because its coefficients have <polynomial growth> while $\phi(n)$ decreases rapidly. Applying the <inverse Fourier transform> gives $u=\sum_n c_ne^{inx}$ in <tempered distributions>. These are the <smooth-window Fourier coefficients of a periodic distribution>, and the transform formula shows independence of $\Psi$.
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