For the Fourier Transform of a distribution ($D'$ space), we have:
If $u\in D'$, and $u$ has compact support, then the Fourier Transform
$$\hat u(\xi)=\langle u(x), e^{-ix\xi}\rangle$$ But since $e^{-ix\xi}$ doesn't have compact support (i.e. not in D space), we understand the RHS term in the sense:
$$\langle u(x), e^{-ix\xi}\rangle = \langle \eta(x)u(x), e^{-ix\xi}\rangle = \langle u(x), \eta(x)e^{-ix\xi}\rangle $$
where $\eta(x)\in D$ and $\eta\equiv1$ in a neighbourhood of $supp(u)$.
My question is that why we need "in a neighbourhood of" instead of just "$\eta=1$ on $supp(u)$"?
And what is the purpose of this $\eta$ ?
Any help or advice is greatly appreciated!