State-space representation of Matérn and Damped Simple Harmonic Oscillator Gaussian processes

Gaussian processes (GPs) are used widely in the analysis of astronomical time series. GPs with rational spectral densities have state-space representations which allow O(n) evaluation of the likelihood. We calculate analytic state space representations for the damped simple harmonic oscillator and the Matérn 1/2, 3/2 and 5/2 processes.