Say I know the behavior of $f:\mathbb{R} \to \mathbb{R}$ in the vicinity of 0. Are there any results linking that to the behavior of its Fourier transform $\widehat{f}(\xi )$ for large values of $|\xi |$?
I did not assume that $f$ belongs to some certain function space on purpose since I do not know when any such result would be valid. As a matter of fact I would mostly be interested in the case when $f\in \mathcal{S}'(\mathbb{R})$ is a tempered distribution. Of course the Riemann-Lebesgue lemma says what happens if $f\in L^1(\mathbb{R})$ regardless of any local description.
EDIT: Let me clarify my question somewhat. What I have is a (kind of) Taylor expansion of $f$ at 0 and I'm wondering if the transformed series says something about the behavior of $\widehat{f}$ at $\infty $?
There are two types of results that I know of, maybe they will satisfy your curiosity, but doubtlessly there are more:
Suppose both $f$ and $\widehat{f}(x)$ are real.
1) Suppose both $f$ and $\widehat{f}(x)$ real. In one direction if $f(x) = 0$ in $(-a,a)$ a small neighborhood of $0$, then $\widehat{f}(x)$ has at least $r a/\pi$ zeros in any interval $[0,r]$ as $r \rightarrow \infty$.
SEE: http://arxiv.org/abs/math/0301060
2) Suppose that $f(x) = 0$ away from a small neighborhood $(-a,a)$ of $0$ Then $\widehat{f}(x)$ is entire, of exponential type, and thus "spreads out" across the entire real line, in particular $f$ cannot decay too fast or that would contradict its analycity.
SEE: This is rather classical. I recommend looking at the "uncertainty principle" and "Paley-Wiener" in any book on harmonic analysis and a book about entire functions such as Levin's "Lecture on entire functions". For a very nice take on the uncertainty principle take a look at http://www.math.msu.edu/~fedja/Published/paper.ps