The Paley-Wiener theorem characterizes the speed of growth of the Fourier transform (say, of exponential type $A$ i.e. of growth essentially bounded by $exp(A \cdot \mathrm{Im}(z))$) in terms of the original function (smooth, supported in $(-A,A)$). In particular, a function compactly supported has its Fourier transform potentially growing exponentially.
I would like to know if there are other theorems of this kind for non-compactly supported functions. In particular : what are the functions $f$ so that its Fourier transform is such that $\hat{f}(x)$ is dominated by $\exp(- A x)$ ?
Thanks in advance, any reference is welcome.