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Suppose we are given an integral operator $ g(x)=f(x)+ \lambda \int_{0}^{\infty}K(x,t)f(t)dt $

with the kernel $ K(x,t)=K(t,x)$. According Hilbert-Schmidt theory then, the function can be obtained as

$ f(x)= \sum_{n=0}^{\infty}\frac{c_{n}}{\lambda _{n} -\lambda}\phi_{n} (x)$

with $ \phi_{n} (x) $ being the eigenfucnctions of the integral kernel $ K(x,t) = K(t,x) $. How can I get these eigenfunctions? Thanks.

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    Again, it depends on what $K$ is. I don't think there's anything reasonable that can be said about the general problem. For an arbitrary $K$ the eigenfunctions should not be expressible in terms of elementary functions, so what does it even mean to know what they are?2012-05-30

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