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
- 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 $
- 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