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
There is a paper related to your question: MR2072747 Ostrovskii, I. V.; Ulanovskii, A. Non-oscillating Paley-Wiener functions. J. Anal. Math. 92 (2004), 211–232.
The Paley-Wiener theorem completely describes the Fourier transforms of functions with bounded support. Because of the importance of these functions in signal processing, and other applications, there is an enormous literature about them.
On your second question. The answer is no. Take a constant function for example. However, if you require that all Fourier coefficients are zero on $(-M,M)$, one can say something on the sign changes, the most comprehensive account of this phenomenon which I know is MR2038065, and a short exposition is MR2048457. These papers are also avilable from the arxiv and from my web page.
By translation and rescaling we can assume that the support is $[-1,1]$. The Paley-Wiener theorem characterizes distributions with support $[-1,1]$. A distribution $u$ is supported in $[-1,1]$ if and only if $\hat u$ is an entire function (holomorphic in $\mathbb C$) such that there exists $M, C$, $$\forall \zeta\in \mathbb C,\quad \vert \hat u(\zeta)\vert\le C(1+\vert\zeta\vert)^M e^{2π\vert\Im \zeta\vert}.\tag{PW} $$ Since it is an iff condition, you can take any entire function $v$ satisfying the estimate that $\hat u$ satisfies in (PW), then its Fourier transform will be supported in $[-1,+1]$: there is no further restrictions.
