Technically, this is not a spectral density but these are the first m coefficients of the low rank approximation for the periodic kernel, which are used in the same way. J is a vector of np.arange(m).

The coefficients of the HSGP approximation for the Periodic kernel are given by:

\[ilde{q}_j^2 = \]
rac{2 ext{I}_j (ell^{-2})}{exp(ell^{-2})}

ilde{q}_0^2 =

rac{ ext{I}_0 (ell^{-2})}{exp(ell^{-2})}

where $ ext{I}_j$ is the modified Bessel function of the first kind.