Can we say something like monotonicity, growth rate and oscillation of the Fourier transform of a banded function $f$ with support $[0, N]$ $$\mathcal{F}f(\xi) = \int_{0}^N f(x)e^{-ix\xi}dx.$$ Of course generally $\mathcal{F}f(\xi)$ is complex. I guess it is impossible for the real part or imaginary part $\mathcal{F}f(\xi)$ to be monotone globally. But is it possible to be monotone on $[0, cN]$? Or have very small oscillation on $[0, cN]$. Here $c$ is a positive constant.

Thanks,

Jack