What are the Eigenvectors of the Integration Operator with Cosine Kernel?

[220205]

Homework Statement

Find the eigenvalues and a corresponding system of eigenvectors of the operator

Af(x) := Integration from 0 to 1 K(x; y)f(y) dy
where
K(x; y) = cos (2pi(x - y))

 

[lanedance]

so you want
Af(x) = \int_0^1 dy \ cos (2 \pi(x - y)) f(y) = \lambda f(x)
what ideas do you have for finding the functions

 

[hunt_mat]

I'm thinking differentiate w.r.t x twice, this should turn it into a differential equation.