Relationship between short-time and large-frequency asymptotics in Fourier transform

Question

I am trying to understand how the short-time behaviour of a function $f(t)$ influences the large-frequency asymptotics of its Fourier transform $g(\omega)=\mathcal{F}[f(t)](\omega)\equiv \tilde{f}(\omega)$, in case it exists. I have read sometimes, and it's apparently very intuitive (not to me), that those limits are directly related, but I don't know how to put this on formal grounds or whether is just loosely stated. As far as I understand, one can define an uncertainty principle relating the characteristic "width" of each function, but this does not tell us anything about the behaviour at large frequencies themselves (in an "absolute" sense). Maybe I am misunderstanding something very basic here.

I came across this doubt while reading this paper, where asymptotic expressions for the short-time limit of the autocorrelation function and the large-frequency behaviour of the power spectrum are derived. At the end of page 13 of the linked pdf file, it is stated:

A small-$\tau$ expansion of $R(\tau)$ from (43) yields $R(\tau)\sim a - 2(1-i\delta)|\tau|+\mathcal{O}(\tau^2,a^{-1})$ as $\tau\rightarrow 0, a\rightarrow \infty$ and the corresponding large-frequency asymptotic form of the spectrum $F(\omega)$ from (20) is $F(\omega)\sim\frac{4}{\omega^2}+\mathcal{O}(\omega^{-3})$ as $\omega\rightarrow \infty$,

where $R(\tau)$ is the autocorrelation function, $a$ and $\delta$ are constants and $F(\omega)=2\Re{\int_0^{\infty}d\tau e^{i\omega\tau}R(\tau)}$ is the power spectrum. The original $R(\tau)$ from which the approximation is derived is

\begin{equation} R(\tau)=a(a_0+a_1 e^{-2a\tau}+a_2 e^{-4a\tau})e^{-D(\tau)}, \end{equation} where $D(\tau)=\frac{1}{a}\tau+\frac{\delta^2}{2a^2}(2a\tau+e^{-2a\tau}-1)$, $a_0=\left( 1+\frac{\delta}{2a^2}i\right)^2$, $a_1=\frac{1}{2a^2}\left( 1+\frac{\delta^2}{a^2}-2\delta i \right)$ and $a_2=-\frac{\delta^2}{2a^4}$.

I understand how to get the approximation for $R(\tau)$, but I don't see how to obtain from there the asymptotics for $F(\omega\rightarrow\infty)$ and how they are related, even though it seems to be trivial for the authors.

To sum up: I'd be interested to know how short-time and large-frequency limit are related (or whether this is only valid e.g. when $f(t)$ is peaked around $t=0$ and there is a characteristic width $\delta t$) and in particular how to obtain the large-frequency asymptotics for the case outlined above.

Thank you very much for your help!

Answer 1

The large frequency asymptotic is related to the points of nonanalyticality of $R(\tau)$. For autocorrelation functions this is typically at $\tau=0$ as the functions are even under $\tau \mapsto -\tau$ but still have a finite derivative...

It is a well-known theorem of Fourier analysis: if a function $R(\tau)$ has $n$ continuous derivatives, its Fourier transform decays like $$F(\omega) \propto \frac1{\omega^{2+n}}.$$

For example $R(\tau) = \exp(-\gamma |\tau|)$ is continuous but has no continuous derivative ($n=0$), and thus we have $F(\omega) \propto \omega^{-2}$.