0
$\begingroup$

Consider the symmetric $L^2$ kernel $K(x,t)=\log(1-\cos(x-t))$ for $0\leq x,t\leq 2\pi$ . Find the eigenvalues and corresponding eigenfunctions of $K(x,t)$

By general procedure we consider a Fredholm equation $$y(x)=\lambda\int_0^{2\pi}\log(1-\cos(x-t))y(t)dt$$ $$\implies y(x)=\lambda\log 2\int_0^{2\pi}y(t)dt+2\lambda\int_0^{2\pi}\log\sin\left(\frac{x-t}{2}\right)y(t)dt$$ $$\implies y'(x)=2\lambda\int_0^{2\pi}\cot\left(\frac{x-t}{2}\right)y(t)dt$$ But in this way I'm unable to convert the equation into a Sturm-Liouville boundary value problem and then find the eigenvalues and eigenfunctions . What is the alternative way to find the same for this problem ? Any help is appreciated .

$\endgroup$
4

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.