Suppose $f(z)$ is a function analytic in the strip $|Re(z)|\leq a$. Is the fourier transform $\hat{f}(w)=o(e^{-a|w|})$? 

It seems plausible but I can't seem to prove it either. 

There is similar result called the Paley-Wiener Theorem that states $e^{a|w|}\hat{f}(w)\in L_2(\mathbb{R})$, but I don't think that helps. 

Thanks in advance.