= Periodic distribution
{title2=$\langle u,\phi(\cdot+T)\rangle=\langle u,\phi\rangle$}
A <distribution> has period $T>0$ if the displayed equality holds for every <test function> $\phi$. This agrees with $f(x+T)=f(x)$ <almost everywhere> for a <regular distribution>. If $u$ is a <tempered distribution>, density of <test functions> in the <Schwartz space> extends the identity to all <Schwartz functions>.
For a <test function> $\Psi$ with $\sum_k\Psi(x+kT)=1$, one has
$$
\langle u,\phi\rangle=\left\langle u,\Psi(x)\sum_k\phi(x+kT)\right\rangle.
$$
Indeed, expand $\phi=\sum_k\Psi(x+kT)\phi(x)$ in the <Schwartz space> and move each translation to $\phi$ by periodicity. All <Schwartz seminorms> of the localized terms decay faster than any power of $|k|$, justifying the sum of pairings. This localization permits rigorous <Fourier coefficients> without multiplying a <distribution> by a discontinuous window.
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