fourier transform - How do I calculate the eigenvalues of a covariance matrix which contains harmonic functions

I have read that $ce^{-\frac{\phi_i-\phi_j}{\rho}}$ is a harmonic function of the form $e^{-in\phi_i}$, and therefore it's eigenvalues are one of the Fourier components of $(|\phi_i-\phi_j|)$.

My questions are:

1. How come $ce^{-\frac{\phi_i-\phi_j}{\rho}}$ has the form of $e^{-in\phi_i}$ if it is a real function?


2. How do I calculate the eigenvalues of such a function, or, if it is a difficult calculation, is there a known theorem(s) I can base this result on?

I am not an expert in harmonic analysis, but I would like to know more about this subject (without getting into too much details). I need it to my research in neurosciences.

Here is a link to the article I am reffering to: Population coding

Thanks!