On the existence of compactly supported functions whose its Fourier transform satisfies a given condition My question is concerned with the existence of compactly supported functions whose its Fourier transform satisfies a given condition: For $\gamma\ge 1$, one can prove that there is no compactly supported function $f$ satisfying $\left| {\hat f\left( t  \right)} \right| = O\left( {{e^{ - k{{\left| t \right|}^\gamma }}}} \right)$ as $|t|\to +\infty$?, for some $k>0$. Here $\hat f$ denotes the usual Fourier transform of $f$. Now I am interested in the case of $0<\gamma<1$. For $\gamma=1/2$, I can give an example of a such function, for example the standard bump function $\varphi \left( x \right) = \exp \left\{ { - \frac{1}{{1 - {x^2}}}} \right\}{\chi _{\left( { - 1,1} \right)}}\left( x \right)$ having $\left| {\hat \varphi\left( t \right)} \right| = O\left( {{e^{ - \sqrt {\left| t \right|} }}} \right)$ (see in http://en.wikipedia.org/wiki/Bump_function). My question is: Given $\gamma\in(0,1)$. Is there a compactly supported function $f$ such that $\left| {\hat f\left( t  \right)} \right| = O\left( {{e^{ - k{{\left| t \right|}^\gamma }}}} \right)$ as $|t|\to +\infty$? 
 A: As Denis Chaperon de Lauzières says, the precise answer to this question is given by the
Beurling-Malliavin theorems. They give a very precise characterization of weights $w\geq 1$
for which there exists a Fourier transform $f$ of a function with bounded support, such that
$f(x)w(x)$ is bounded.
The necessary condition is that
$$\int \frac{\log w(x)}{1+x^2}dx<\infty.$$
This is the famous "Logarithmic integral".
To make it also sufficient one needs some regularity. For example, if $\log w$ is uniformly
continuous then this is also sufficient. Another regularity condition is that $w$ is
itself a functon of exponential type with
$$\int\frac{\log^+|w(x)|}{1+x^2}dx<\infty.$$
References: Beurling-Malliavin, Acta math. 107 (1962) 291-309,
P. Koosis, Logarithmic Integral,
P. Koosis, Lecons sur les theoremes de Beurling-Malliavin.
A: The oldest reference I know which constructs compactly-supported functions with as good as possible decay of the Fourier transform at infinity is A. Ingham, "A note on Fourier transforms", J. London Math. Soc. s1–9 (1934), 29–32. 
See also Th. 1.3.5 in Hörmander's "The analysis of linear partial differential operators I: distribution theory and Fourier analysis", Springer-Verlag, 2003.
(I think more refined questions are addressed by the "Beurling - Malliavin multiplier theorem").
A: Yes there is. What you need is the Paley-Wiener Theorem as in http://en.wikipedia.org/wiki/Paley–Wiener_theorem
