How will covariance matrix converge to the covariance function, especially in the aspect of eigenvalues and eigenvectors

Question

In spatial statistics, people want to estimate the covariance function $C(x,y)$, $x,y\in [0,1]$. Because this covariance function is positive definite, we have the following decomposition $$C(x,y)=\sum_{i=1}^\infty\lambda_i \nu_i(x)\nu_i(y),$$ where $\lambda_i$ and $\nu_i$ are eigenvalues and eigenfunctions separately. However, most of the time we could only estimate a covariance matrix $C_n=(C(\frac{j}{n}),C(\frac{k}{n}))_{j,k},j,k=1,\dots,n$, where $n$ is the number of grids. And then we calculate the eigenvalues and eigenvectors of $C_n$, let's say they are $\lambda_{n,i}$ and $\nu_{n,i}$. I was wondering under what kind of norm and what kind of rate $\lambda_{n,i}$ and $\nu_{n,i}$ will converge to $\lambda_i$ and $\nu_i$. You may add some common conditions if you want, for smoothness or decreasing rate of $\lambda_i$. I hope you could provide concise proof or inference. Thanks a lot!!