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,  

$$\lim\_{N\rightarrow \infty} \int \int |K(x,y) - \sum\_{i=1}^N \lambda\_i e^i(x)e^i(y)|^2 d\mu(x)d\mu(y) = 0.$$



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> 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.. 
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Error estimate in the spectral theorem of compact operators on a Hilbert space
I'm following [Szego's book on orthogonal polynomials](https://books.google.co.il/books?hl=en&lr=&id=RemVAwAAQBAJ&oi=fnd&pg=PR9&dq=szego%20orthogonal%20polynomials&ots=l4bK485fTQ&sig=S6hN5M_fZqiHhr2ny7sOcV0oSpU&redir_esc=y#v=onepage&q=szego%20orthogonal%20polynomials&f=false).


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)](http://www.jstor.org/stable/pdf/1990404.pdf?_=1461643206256), 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.


