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.