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

Question

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!

Answer 1

1. 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).

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