Consider the following exponential kernel:

$k(x_1, x_2) = \exp\left(\frac{|x_1 - x_2|}{L}\right)$,

which is symmetric and non-negative definite. By virtue of Mercer's theorem, we have

$k(x_1, x_2) = \sum_{i = 1}^\infty \lambda_i \phi_i(x_1) \phi_i(x_2)$

where $\lambda_i$ and $\phi_i$ are the eigenvalues and eigenfunctions of $k$, respectively. Now, consider the following product:

$K((x_1, y_1), (x_2, y_2)) := k(x_1, x_2) k(y_1, y_2) = \exp\left( -\frac{|x_1 - x_2|}{L} - \frac{|y_1 - y_2|}{L}\right)$.

Since the product of two symmetric, non-negative definite kernels is another kernel with the same properties, Mercer's theorem still applies.

The question is: Having computed $\lambda_i$ and $\phi_i$ of $k$, what can we say about the eigenfunctions and eigenvalues of $K$?

Thank you.

Regards, Ivan