Let $f$ be a smooth function on $\mathbb{R}^n$ with compact support and let $g : \mathbb{R}^n \rightarrow \mathbb{R}$ be a smooth function wich vanishes on a set $A$. We consider another smooth functions $ \chi$ wich vanishes on the set $A$.
My question does the following function converge when $a \rightarrow + \infty$
$$ \chi \int_0^a t^k \hat{f}(-tg)dt,$$
Where $\hat{f}$ is the Fourier transform of $f$ ?