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The following is from Stein and Shakarchi's Complex Analysis:

For each $a>0$ we denote by ${\mathcal F}_a$ the class of all functions $f$ that satisfy the following two conditions:

  1. The function $f$ is holomorphic in the horizontal strip $$ S_a=\{z\in{\Bbb C}:|Im(z)|<a\} $$
  2. There exists a constant $A>0$ such that $$ |f(x+iy)|\leq\frac{A}{1+x^2}\quad\text{for all}\quad x\in{\Bbb R}, |y|<a. $$

Denote by ${\mathcal F}$ the class of all functions that belong to ${\mathcal F}_a$ for some $a$. The the Fourier inversion holds in this class.

Here are my questions:

  • Is there a name for this class?
  • Does it have anything to do with the Schwartz space on which the Fourier transform is a linear isomorphism?
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  • $\begingroup$ I don't think there's a canonical name for this class, nor any relation to the Schwartz space. $\endgroup$ – Potato Feb 21 '13 at 1:28
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For the sake of an answer, this is done on MO.

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