Given a function $f(x)$ and its Fourier transform $\tilde{f}(k) = \int_{-\infty}^{\infty}f(x) e^{ikx}dx$, if I decompose the Fourier transform as $$\tilde{F}(k) = A(k)e^{ikl(k)}$$ Under what conditions on $f(x)$ can I say that $l(k)$ is a slowly varying phase factor ($\dfrac{d l}{dk}$ is small), i.e. when does the Fourier transform look approximately sinusoidal $\tilde{f}(k) \sim \sin(kd + \phi)$?
For example, the functions $\dfrac{\sin kd}{kd}$, which is the Fourier transform of a box centered at the origin and $\cos(kd) e^{-k^2}$, the Fourier transform of a symmetric shifted gaussian both are acceptable. In both these cases, $f(x)$ has a finite support ($f(x) = 0$ for $|x| >$ some L). Does having a finite support help in general?
