Can limits of be computed via Fourier transforms

Question

Suppose, we want to find a limit of $f(t)$ or $t^n f(t)$ as $t \to \infty$. Can these limits be computed via fourier tranform of $f(t)$?

Answer 1

A function $f$ is absolutely continuous on $[a,b]$ iff there is a function $g$ that is absolutely integrable on $[a,b]$ and satisfies $$ f(y)-f(x) = \int_{x}^{y} g(t)dt,\;\;\; x,y \in [a,b]. $$ Equivalently, a function $f$ is absolutely continuous on $[a,b]$ if it is continuous on $[a,b]$, has a derivative a.e. on $[a,b]$, the derivative is integrable on $[a,b]$ and $f(y)-f(x)=\int_{x}^{y}f'(t)dt$ for all $x,y\in [a,b]$.

The Fourier transform is perfectly tied to the differentiation operator as follows: $f \in L^2(\mathbb{R})$ is absolutely continuous on every finite interval, with $f' \in L^2(\mathbb{R})$ iff $$ \int_{-\infty}^{\infty}|\hat{f}(\xi)|^2+|\xi\hat{f}(\xi)|^2d\xi < \infty. $$ And $f$ is n-times absolutely continuous with $f,f',\cdots,f^{(n)}\in L^2$ iff $$ \int_{-\infty}^{\infty}|\hat{f}(\xi)|^2+|\xi\hat{f}(\xi)|^2+\cdots+|\xi^{n}\hat{f}(\xi)|^{2}d\xi < \infty,\\ \mbox{ or } \int_{-\infty}^{\infty}(1+|\xi|^{2n})|\hat{f}(\xi)|^2d\xi < \infty. $$ That's about it when it comes to iff conditions.