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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.

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1 Answer 1

up vote 7 down vote accepted

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)).

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In this book Theorem IX.14 is the one I want! Thanks! The proof isn't given though... would you have any suggests on where to look. Thanks!! :) –  alext87 Oct 30 '10 at 17:35
    
Hmm... A short outline of the proof is given in Problem 76 in Chapter IX. I will update my answer as soon as I find a better reference. –  Andrey Rekalo Oct 30 '10 at 18:02
    
I like the book by Koosis, Introduction to $H_p$ Spaces. The one I have is the second edition by Cambridge University Press from 1998 which definitely does have a nice discussion on the Paley-Wiener Theorem (and many other topics), which should be adaptable to this without too much difficulty (it might even show this exactly, I'll have to check). It needs a little bit of complex analysis, i.e. a Phragmen-Lindelof theorem or similar. –  Zen Harper Dec 3 '10 at 10:45

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