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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?

Random function and it's Fourier transform

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Since your question is not very rigorous I'll give the simplest case:

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.

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