In a class about statistical learning, the professor was talking about various theorems like Mercer's theorem, Bochner's theorem, and representer theorem in context of Gaussian Processes (GPs). Two phrases that the instructor used that confused me were "data sampled from a GP" and the "spectral representation of a GP," where for the latter he referred to Bochner's and Mercer's theorem. So I have two related questions.
My understanding is that a GP is a stochastic process such that any two random variable $f(x_1), f(x_2)$ have means given by $m(f(x_i))$ and covariance by $K(f(x_1), f(x_2))$, and inductively any finite collections are distributed multivariate normal (MVN). Given this definition of a GP, what exactly does "data sampled from a GP" mean? If a GP is an infinite collection of random variables, does it simply mean a finite number of "crystallization" of those random variables? Or if one sees a GP as a distribution over function space, is a "sample" from a GP, then a random function?
What does Bochner's theorem have to do with a "spectral representation of a GP"? Spectral representations usually refer to Fourier transforms of measurable functions, or eigendecompositions of matrices. A GP is neither of these, so how can it have a "spectral representation"? I understand how Mercer's theorem is the "infinite" analog of matrix diagonalization for an operator, but I don't see how Bochner's theorem says anything about spectra, for spectra as defined above? Intuition and simple explanation would be appreciated.
Thank you so much!