Error estimate in the spectral theorem of compact operators on a Hilbert space Given a compact self-adjoint operator $K$ mapping $L^2(\mathbb{R}^d) \rightarrow  L^2(\mathbb{R}^d)$ as $f \rightarrow \int K(x,y) f(y) d\mu(y)$,  let us define its eigenvalues $\lambda_i$ and eigen-functions $e^i(x)$. 
Let us 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.. 

 A: I'm following Szego's book on orthogonal polynomials.
In chapter III he considers $f \in L^2\left((a,b),d\alpha \right)$, with $-\infty \leq a < b \leq \infty $. There exist a set of orthogonal polynomials $\{p_n \}_{n=0}^{\infty}$ w.r. to the measure $d\alpha$. 
We consider the kernel $K_n(t,x) := \sum\limits_{\nu=0}^{n} \bar{p_{\nu} (t)} p_{\nu} (x) $. Then one can show that $\int\limits_{a}^{b} K_n (t,x) f(t) \, d\alpha (t) = \sum\limits_{\nu = 0}^{n} \hat{f}(n)p_n (x) $, the spectral expension of order $n$. 
So, if we settled that the polynomial expansion of $f$ is an integral transform, there's a detailed discussion in Szego's book, Davis "approximation theory" and many other books about its $L^2$ error.
Edit: If we look in Aronszajn (1950), we can have by Section 9 that the kernels converge in L^2 norm.
Edit 2: I recommend you to read the aforementioned paper, as it deals with the much broader class of reproducing kernels, and give a lot of conditions and results that are relevant to your question.
A: Actually, the presumption that the partial sums of the kernel converge to it in $L^2$ of the product is not quite right: for example, mapping $\ell^2\to \ell^2$ by $e_n\to \lambda_n\cdot e_n$ for a sequence of real numbers $\lambda_n$ going to $0$, but slowly, gives kernel $K=\sum_n \lambda_n\cdot e_n\otimes e_n$. For $\lambda_n$ going to $0$ slowly enough, certainly $\sum_n |\lambda_n|^2=+\infty$, so the kernel cannot be in $L^2$ of the product. This obstacle exists prior to talking about spaces of functions on reasonable physical spaces.
In general, asking about pointwise (much less uniform pointwise) convergence is somewhat worse, when the Hilbert spaces are concrete spaces of functions.
So maybe this is not quite an "answer to the question", but a reaction and request for clarification.
