Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
    
It depends on what $K$ is! The eigenvalue equation is an equation and you have to solve it. –  Qiaochu Yuan May 30 '12 at 18:43
    
ok but how about the eigenfunctions ?? –  Jose Garcia May 30 '12 at 18:48
    
You simply have to find those too. I don't really understand what you're asking. Of course in the finite-dimensional case you can compute a determinant to find the eigenvalues and then row reduce to find their eigenvectors but in the infinite-dimensional case you really just have to solve the equation! Is there a particular $K$ you're struggling with? –  Qiaochu Yuan May 30 '12 at 18:51
    
isn't there a general method to get eigenfunctions ??, i have searched many books of integral equation and found little info about it –  Jose Garcia May 30 '12 at 20:05
    
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? –  Qiaochu Yuan May 30 '12 at 20:06

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.