How to show something is a convolution operator?

Question

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 $L^{\infty}$.

Answer 1

Partial result: $$W(a) W(b) = F^{-1}a\underline{F\ F^{-1}}bF = F^{-1}abF = W(ab)$$ where the underline just shows what is "obviously" going to cancel.

Answer 2

In general, the Fourier transform of a product of two functions is the convolution of those functions: see here. So

$$W(a) u = F^{-1}(a \hat{u}) = \hat{a} \star u $$