I have the operator $W(a)$ defined by $$W(a)=F^{-1}aF$$ where $F$ denotes the fourier transform and $a$ is a function on $L^{\infty}$. I need to prove that this is convolution operator, but I don't know how.
Also how can you prove the following property of $W(a)$: $$W(ab)=W(a)W(b)$$ for any functions $a,b$ in $L^{\infty}$.