I'm trying to determine the nullspace and range of the following integral operator, but I'm having trouble proceeding. Let $K:C([0,1])\to C([0,1])$ be defined by $$Kf(y)=\int_{0}^1 \sin(\pi(x-y))f(y)\,dy.$$ Playing around with several functions, I see that if $f\equiv 1$, then $$K(y)=\int_{0}^1\sin(\pi(x-y))\,dy=-\frac{2\cos(\pi x)}{\pi},$$ and if $f\equiv 0$, then $Kf(y)=0$.
The form $\sin(\pi(x-y))$ in $K$ makes me think that the range might be periodic function in $C([0,1]), but this is a guess with no intuition.
Edit: I didn't include this originally, but as Daniel's mentioned, the addition formula implies that $$(Kf)(y)=\sin (\pi x) \int_{0}^1 \cos (\pi y)f(y)\,dy - \cos (\pi x) \int_{0}^1\sin (\pi y)f(y)\,dy.$$ However, I'm not seeing what this implies either. In this form $K$ reminds me of the Riemann-Lesbegue lemma for $L^1([0,1]),$ but I'm not sure what what that invokes.
So my question is ultimately: what should I look for when determining the range and nullspace of integral operators?