TITLE: Error estimate in the spectral theorem of compact operators on a Hilbert space
QUESTION [5 upvotes]: 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..

REPLY [2 votes]: 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.