Let $g(x)$ be smooth and compactly supported on $\mathbb{R}^+$. I know the Fourier transform $\hat{g}(y)$ concentrates on a short segment near $y=0$ if the support is big. However, I'd like to see how "short" this segment is in terms of $g(x)$ and the length of supp $g$.
I got some rough bounds on $\hat{g}(y)$ with some extra conditions such as the growth of $g^{(j)}$ using integration by parts but I don't see how to show this in general. Can anyone help me with it?