= Solution
A <distribution> $u$ is a $2\pi$ <periodic distribution> when
$$
\boxed{\langle u,\theta(\cdot+2\pi)\rangle=\langle u,\theta\rangle\qquad(\theta\in C_c^\infty(\mathbb R)).}
$$
Iteration gives invariance under all integer multiples of $2\pi$. For a <tempered distribution> the identity also holds for every <Schwartz function>, since <test functions> are dense in the <Schwartz space>: multiplying by <cutoff functions> $\chi(x/R)$ equal to one near zero gives such an approximation in every <Schwartz seminorm>.
For $\phi\in\mathcal S(\mathbb R)$ define $a_k(x)=\Psi(x+2\pi k)\phi(x)$. The <smooth periodic partition of unity> gives $\phi=\sum_ka_k$ with convergence in the <Schwartz space>. Indeed, each $a_k$ is supported where $|x+2\pi k|\leq2\pi$, and the <generalized Leibniz rule> and arbitrarily rapid decay of the <derivatives> of $\phi$ give, for every $m,n,M$,
$$
p_{m,n}(a_k)\leq C_{m,n,M,\phi}(1+|k|)^{-M}.
$$
Consequently continuity and periodicity of the <tempered distribution> give
$$
\begin{aligned}
\langle u,\phi\rangle
&=\sum_k\langle u,\Psi(\cdot+2\pi k)\phi\rangle\\
&=\sum_k\langle u,\Psi(\cdot)\phi(\cdot-2\pi k)\rangle\\
&=\left\langle u,\Psi(\cdot)\sum_k\phi(\cdot-2\pi k)\right\rangle
=\boxed{\langle u,\Psi P\phi\rangle}.
\end{aligned}
$$
For the last interchange, all summands have the fixed <compact support> of $\Psi$ and their <derivatives> decrease faster than any power of $|k|$. Hence the series converges in the <space of test functions>, as well as in the <Schwartz space>. Notice that $P\phi$ itself need not be a <Schwartz function>; the factor $\Psi$ makes the pairing legitimate.
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