Show that the quadratic function

\[
\frac{2 \pi^{2}}{3}-x(2 \pi-x)
\]

on \([0,2 \pi)\) has Fourier cosine coefficients \(4 \pi / k^{2}\) for \(k \geq 1\). Hence or otherwise, investigate the function

\[
K\left(t, t^{\prime}ight)=\frac{2 \pi^{2}}{3}-\left|t-t^{\prime}ight|\left(2 \pi-\left|t-t^{\prime}ight|ight)
\]

as a candidate covariance function for a process on \([0,2 \pi)\), and by extension to a stationary periodic process on the real line. Plot a simulation of the process on \([0,4 \pi)\), and verify continuity.