I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple and provide me some references.

What is the eigenfunction of a multivariate Gaussian kernel: \begin{equation} f(x,y) = \exp\left(-\frac{\lVert x - y\rVert^2}{2\sigma^2}\right) \end{equation}

I am interested in the eigenfunctions with respect to $L^2$ norm: \begin{equation} \int f(x, y) v_i(y) dy = \lambda_i v_i(x) \end{equation} and also with respect to a Gaussian probability distribution $p(x)$: \begin{equation} \int f(x, y) v_i(y) p(y) dy = \lambda_i v_i(x) \end{equation} I am aware of the numerical approximation to the problem like Nystrom method. However, it should be possible to find a closed form solution when the probability distribution is also Gaussian.