Given $g:[1,\infty) \rightarrow (0,\infty)$ with $g(t) = o(t)$, does there exist $f:\mathbb{R} \rightarrow [0,\infty)$ with support contained in $[-1,1]$ such that $$ \widehat{f}(y) = \int_{\mathbb{R}} f(x)\exp(-2\pi i x y) dx = O(\exp(-g(|y|)))? $$
Motivation:
If $f$ is compactly supported and $\widehat{f}(y) = O(\exp(-c|y|))$, then $f$ is everywhere $0$. (See the question Compactly supported function whose Fourier transform decays exponentially?.)
The function $$ f = \left\{ \begin{array}{ll} \exp(-1/(1-x^2)) & \text{if } x\in (-1,1) \\ 0 & \text{otherwise} \end{array}\right. $$ has $\widehat{f}(y) = O(\exp(-\sqrt{|y|}))$. (See http://math.mit.edu/~stevenj/bump-saddle.pdf. By the way, does anyone have a textbook or published reference as an alternative to this note?)
