Let $k\colon\mathbb{R}\to\mathbb{C}$ be a 1-periodic function with $k|[0,1]\in L^2([0,1])$. Define the convolution operator $T$ as $f\mapsto\int\limits_{[0,1]}k(s-t)f(t)\, dt$. Develop $k$ in a Fourierseries and with this find the eigenvalues and eigenfunctions.
Could anyone please help me? I never did something like this before.
I guess the development of k in a Fourierseries is
$k(t)=\frac{a_0}{2}+\sum\limits_{k=1}^{\infty}(a_k\cos(k2\pi t)+b_k\sin(k2\pi t))$.
And what is the next step?