= Solution
Let $D=\sum_{k\in\mathbb Z}\delta_{2\pi k}$ be the <Dirac comb>. Pair the given identity $D=(2\pi)^{-1}\sum_{n\in\mathbb Z}e^{in\cdot}$ with the <Schwartz function> $y\mapsto\phi(x-y)$. On the left this gives $\sum_k\phi(x-2\pi k)=P\phi(x)$. On the right,
$$
\int_{\mathbb R}e^{iny}\phi(x-y)\,dy
=e^{inx}\int_{\mathbb R}\phi(z)e^{-inz}\,dz
=e^{inx}\widehat\phi(n).
$$
Therefore the <Fourier series> is
$$
\boxed{P\phi(x)=\frac1{2\pi}\sum_{n\in\mathbb Z}\widehat\phi(n)e^{inx}.}
$$
The <Fourier bounds for Schwartz functions> imply $|\widehat\phi(n)|\leq C_N(1+|n|)^{-N}$ for every $N$. For each nonnegative integer $j$, choose $N>j+1$ to obtain $\sum_n|n|^j|\widehat\phi(n)|<\infty$. Thus the <Fourier series> and every differentiated <Fourier series> converge absolutely and uniformly on all of $\mathbb R$. In particular convergence holds in the topology of <smooth> $2\pi$ <periodic functions>, defined by uniform convergence of every <derivative> on one period, and also in <tempered distributions>.
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