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 $a$. Then the Fourier inversion holds in this class.

My questions are: 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?