In Proposition 5.3.11 of Bratteli-Robinson Vol II the Paley-Wiener Theorem is stated as follows:

Excerpt from Bratteli-Robinson

Is this the correct statement? Aren't $f$ and $\hat{f}$ mixed up? I know the following version of this theorem (see for example Hörmander Vol I, Theorem 7.3.1):

If f is in $C_0^\infty(\mathbb{R})$ with support in $[-R,R]$, then $\hat{f}$ can be extended to an entire analytic Funktion satisfying that for all $n\in\mathbb{N}$ exist constants $c_n$ such that $\lvert \hat{f}(z)\lvert\leq c_n\frac{e^{R\lvert\Im{z}\lvert}}{(1+\lvert z\lvert)^n}$. Conversely an entire analytic function $\hat{f}$ satisfying this inequality for all $n$ is the Fourier transform of some $F$ in $C_0^\infty(\mathbb{R})$ with support in $[-R,R]$.

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    $\begingroup$ I think they're saying the same thing since $f^\vee(z) = \hat{f}(-z)$. $\endgroup$ – Suugaku Jun 2 '13 at 14:43
  • $\begingroup$ @Suugaku But one is the Fourier transform, the other the inverse transform. Why are they the same? $\endgroup$ – awllower Jun 2 '13 at 14:52
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    $\begingroup$ @awllower I'm saying it doesn't matter whether it says FT or IFT since the properties do not rely on whether $z$ or $-z$ is used. The answer below sums this up. $\endgroup$ – Suugaku Jun 2 '13 at 21:30
  • $\begingroup$ @Suugaku Thanks for telling me. $\endgroup$ – awllower Jun 3 '13 at 0:57

All the involved properties (P) are invariant with respect to $z\mapsto -z$. Hence to be the Fourier transform of a function satisfying (P) is the same as to be the inverse Fourier transform of a function satisfying (P).


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