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. 



Decay of the Fourier transform 
This is not true without additional integrability conditions on $f(\cdot+iy)$. 


$\hat{f}(w)=o(e^{-a|w|})$ implies that $e^{b|w|}\hat{f}(w)\in L\_2(\mathbb{R})$ for all $b < a$. The latter inclusion holds if and only if $f(z)$ is analytic in the strip $|\Im(z)|< a$ *and* $$ \sup\limits\_{|y|\leq b}\|f(\cdot+iy)\|\_{L^2(\mathbb R)}<\infty$$
for all $b< a$ (see *Fourier Analysis, Self-Adjointness* by Reed and Simon (Methods of Modern Mathematical Physics, Vol. 2, Theorem IX.13)).


