I am reading Terry's blogpost about the uncertainty principle
https://terrytao.wordpress.com/2010/06/25/the-uncertainty-principle/
In it I quote the heuristic we often practically use:
"The behaviour of a function $f$ at physical scales above (resp. below) a certain scale $R$ is almost completely controlled by the behaviour of its Fourier transform $\hat f$ at frequency scales below (resp. above) the dual scale $1/R$ "
I would like to understand better how to interpret this best.
Here is my interpretation:
If we consider $f(x)-f(x+t)$ for $t = \Theta (R)$, then by the fourier inversion formula, if we consider the frequencies above $1/R$, they will tend to cancel out since they're spinning fast and randomly. Thus we say this difference depends on $\hat f$ in $[-1/R,1/R]$. However we can do even better- for very small $\epsilon$, as a function of $s$, $\hat f(\epsilon)e^{i\epsilon s}$ doesn't change much when $s$ moves from $x$ to $x+t$. Thus the heuristic I offer is that changes at scale $R$, i.e $|f(x)-f(x+t)|$ for $t = \Theta (R)$, doesn't depend on higher frequencies since they self cancel, and depends on $\hat f$ in $[-1/R,1/R]$ in a weighted form, where the edges have more weight, and as frequency approaches $0$ it has no effect (notice that very high and very low frequencies don't matter for DIFFERENT REASONS).
We can then see this happening in examples; if you take a band-limited function, then on fine scales of $\epsilon$, it depends mostly on $\hat f(1/\epsilon)$ which we assumed is $0$, so it is very smooth (and the band-limited in the middle has barely any effect).
Is this a good interpretation? I get the vibe terry likes to look at the whole interval and not weight different parts of it.