= Periodic covariance function
{title2=$k_P(t,t')=A^2\exp[-2\ell^{-2}\sin^2(\pi(t-t')/P)]$}
This <covariance function> is periodic with period $P>0$. Map time to the circle $u(t)=(\cos(2\pi t/P),\sin(2\pi t/P))$ and pull back a <Gaussian kernel>; since $\|u(t)-u(t')\|^2=4\sin^2(\pi(t-t')/P)$, this proves positive semidefiniteness. A <Gaussian process> with this kernel has $\operatorname{Var}(f(t+P)-f(t))=0$ and therefore repeats at every fixed phase. Replacing the inner $\pi$ by $2\pi$ changes the actual period to $P/2$.
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