# Approximately sinusoidally varying Fourier transforms

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 length scale ($$\dfrac{d l}{dk}$$ 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? I've tried fitting cubic splines to random points and almost always, my Fourier transform is sinusoidal with some decaying envelop (I've attached a picture of one such example). Is this something general?

If $$\|k \hat{h}(k)\|_{L^1}$$ is small then $$\|h'\|_\infty\le \|k \hat{h}(k)\|_{L^1}$$ is small so that for $$|k_0|$$ large enough
$$F(x)=h(x)e^{i k_0 x}$$ can be seen as a complex sine with slowly varying amplitude/phase.
$$\|k \hat{h}(k)\|_{L^1} = \|(k-k_0) \hat{F}(k-k_0)\|_{L^1}$$
The large enough is a bit hard to quantify because it depends on how small $$h(x)$$ can be. If $$h$$ is compactly supported on $$[a,b]$$ then looking at $$\frac{\|k \hat{h}(k)\|_{L^1}}{\|\hat{h}(k)\|_{L^1}}(b-a)$$ is fine.