Eigenvalues of operator between L^2

[Raven2816]

Homework Statement

>M: L_2 -> L_2
>
>(Mf)(t) = int(-pi, pi) sin(y-x)f(x) dx
>
>how do i find eigenvalues/vectors of M and what can i use to find
>information about the spectrum?

Homework Equations

The Attempt at a Solution

now i know that sin(y-x) = sinycosx-cosysinx
i also realize that the range is 2dimensional
when i went to construct f i made it = cosy + asinx
so i plugged this in, but when i integrated i got 0. did i integrate wrong? or did i take the wrong approach?

 

[maverick280857]

Can you give the steps that led you to the particular form of f(x) that you have stated?

 

[Raven2816]

because i know that the eigenvector is contained in a subspace spanned by sin(x) and cos(x). i figured that'd be a good f?