One-sided spectral density of a real stationary time series (${\gamma }_k={\int }_0^{\pi }f(\omega )\cos (k\omega )d\omega$)

For real stationary processes, the one-sided density on $[0,\pi ]$ is twice the usual two-sided density. Its integral is the variance and ${\gamma }_k={\int }_0^{\pi }f(\omega )\cos (k\omega )d\omega$. When covariances are absolutely summable, $f(\omega )={\pi }^{-1}({\gamma }_0+2{\sum }_{k\ge 1}{\gamma }_k\cos (k\omega ))$.