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I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple and provide me some references.

I've and Kernel function $K(x,y)$
$f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$
$\Omega$, a compact set of $\Bbb R^2$. Let's assume that $K$ maps from $L^2(\Omega)\rightarrow L^2(\Omega) $ and it has an orthonormal basis of eigenpairs$(\lambda_i,f_i)_{i \in N }$. My question is: is there any general theory regarding the nature of $K(x,y)$ in the following cases

If I need all the eigenfunctions $(f_i(x))$ such that
$\partial_{x_1} f_i(x)=\partial_{x_2} f_i(x) , \forall i \in \Bbb N $
If any general theory does not exist then can someone give me any idea how to construct a $K(x,y)$ satisfying above ?

:Arn:

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    $\begingroup$ Isn't $f$ a function of one real variable $x$? $\endgroup$
    – Pietro Majer
    Commented Sep 7, 2012 at 7:48
  • $\begingroup$ No $f: \Omega \rightarrow R $ with $\Omega \subset R^2 $ $\endgroup$
    – user26265
    Commented Sep 7, 2012 at 17:07

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