First of all, I do apologise if the question is not formulated in precise mathematical terms, but as a physics student I lack a formal background on rigorous functional analysis.

Suppose we have a Fourier integral of the type

\begin{equation} F(\omega)=\int_{-\infty}^{\infty} dt e^{-i\omega t}f(t) \end{equation}

and we want to obtain an asymptotic expansion as $\omega\rightarrow 0$ that approximates the behaviour of $F(\omega)$ at low frequencies. The function $f(t)$ decays rapidly enough as $t\rightarrow \infty$ so that the previous integral exists.

My question is: is there any general approach to obtain such an asymptotic expansion as $\omega\rightarrow 0$ that goes beyond the leading term (setting $\omega=0$ and trying to perform the integral)?

My interest comes in basically since all the literature I looked through on asymptotic expansions of integrals ("Advanced Mathematical Methods for Scientists and Engineers" by Bender&Orszag; "Applied Asymptotic Analysis" by P.D. Miller, and some lecture notes I found online) focuses only on the high-frequency limit, i.e. as $\omega\rightarrow \infty$, for which methods like steepest-descent or stationary phase are available (or in Laplace-type integrals, Laplace's method).

My guess is that it'd be possible to expand the exponential in powers of $\omega t$ and try to perform the integral term by term if the integral boundaries were finite (with better results if $f(t)$ decays quickly to 0), but I'm not completely sure about that either.

Note: even though I'm interested in the general case, to give a bit of background: $f(t)$ in my case would be an autocorrelation function $C(\tau)=<x(t_0)x(t_0+\tau)> - <x(t_0)>^2$ of a stationary stochastic point process $x(t)$; so $F(\omega)\equiv S(\omega)$ is the power spectral density of the process and the previous statement is equivalent to the Wiener-Khinchin theorem. $C(t)$ and $S(\omega)$ are even functions in $t$ and $\omega$ respectively.

Thank you in advance!


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