Given a compact self-adjoint operator $K$ mapping $L^2(\mathbb{R}^d) \rightarrow  L^2(\mathbb{R}^d)$ as $f \rightarrow \int K(x,y) f(y) d\mu(y)$  lets define its eigenvalues $\lambda_i$ and eigen-functions $e^i(x)$. 

Lets define a sequence of "partial sum" operators $K_N(x,y) = \sum_{i=1}^N \lambda_i e^i(x)e^i(y)$. Then this sequence converges to $K(x,y)$ in the product Hilbert space under the product measure.  

More explicitly then,  

$lim_{N\rightarrow \infty} \int \int |K(x,y) - \sum_{i=1}^N \lambda_i e^i(x)e^i(y)|^2 d\mu(x)d\mu(y) = 0$ 

- But is there any example where for some specific $K$ someone has tried estimating this error-integrand $|K(x,y) - \sum_{i=1}^N \lambda_i e^i(x)e^i(y)|^2 $ ? I would be hapy to find any reference along these lines..