How do I calculate the eigenvalues of a covariance matrix which contains harmonic functions
The included document does not say $ce^{-\frac{\phi_i-\phi_j}{\rho}}$ has the form of $e^{-in\phi_i}$. The included document uses $C$ for three completely different things: 


$C_{ij}$ is the correlations between $i$ and $j$ items.
$C(\phi_i-\phi_j)$ a component of that correlation, dependent on the angles between the two items $i$ and $j$.
$C_n$ the eigenvalues of the covariance matrix made up of the $C_{ij}$.



The term $e^{-in\phi_i}$ is a component of $C_n$ the eigenvalues / Fourier components expression in (31).


The idea would be to form the matrix $C_{ij}$ and then take the eigensystem decomposition of that matrix.