# Properties of Eigenfunctions of a Kernel

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 one Hilbert space to another; let's say $L^2(\Omega)$ 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

1. If I need all the eigenfunctions $(f_i(x))$ such that for all $f_i$ the partial derivatives of $f_i$ wrt $x_1$ and $x_2$ will be same.$\partial_{x_1} f_i(x)=\partial_{x_2} f_i(x) , \forall i \in \Bbb N$
2. This's simple. I only need the Eigenfunctions to have nice regularity. Here I need only some reference in some books or papers.

Lastly I've tried here writing Latex but it seems the same code as it is does not work here.Any tricks?

Arwin

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$\LaTeX$ can be added by putting your math notation between dollar signs $...$. For displayed equations use $$...$$. There's a short guide here. –  axblount Sep 6 '12 at 19:26
The derivative condition in 1 says that $f_i$ is constant on every line with slope $-1$. Any linear combination of such functions will have the same property; therefore they cannot form a basis of $L^2$. –  user31373 Sep 8 '12 at 3:58