I think this question isn't that hard, but I am a bit confused:
Define $(Af)(x):=\int_{0}^{1}\cos(2\pi(x-y))f(y)dy.$ Then $A$ is an operator on functions. Find the eigenvalues and the eigenfunctions.
I can think of a lot of functions that give $0$, things like $f(x)=\cos(n2\pi x)$. Also one eigenfunction that gives eigenvalue $\frac {1}{2}$ (I think). My problem is I have no idea how to show that I have found all of them, and I don't know if I have found all of them.
Thanks for showing me!!
Edit: I havent really shown my work because there is a lot of it and it is all over the place. J.M. suggest that it is useful to write $Af(x)=\cos(2\pi x)\int_{0}^{1}\cos\left(2\pi y\right)f(y)dy+\sin\left(2\pi x\right)\int_{0}^{1}\sin\left(2\pi y\right)f(y)dy.$ I used this to find that $\cos(2\pi x)+\sin (2\pi x)$ is an eigenvector. I wasn't sure if this is the right track, and I am still confused about what to do from here. (Specifically, once we find a bunch of $\lambda$ how do we prove that is all of them?)