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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 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

share|improve this question
    
$\LaTeX$ 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

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