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?



Class of functions in which the Fourier inversion holds
This space has no name (no that I know), and it is probably invented by Stein and Shakarchi for
some pedagogical purpose, or for convenience of exposition.


As Serge says it is not contained in the Schwartz space $S$ (of test functions). Neither it contains the Schwartz space $S$.


It is contained in the space of Schwartz distributions $S^\prime$ (tempered) but probably the authors did not
want to introduce distributions in the elementary Complex Analysis textbook.
As it contains all examples they want to consider in this book, they introduced it.


There are several other spaces where Fourier transform can be defined, and even the image can be
explicitly described, for example, $L^2$, or more general distributions than Schwartz,
but I agree with Bazin that Schwartz distributions is perhaps the
most useful class.


In an elementary textbook, one has to begin with something, so they invented this artificial class.
In the later volumes of the course they address $L^2$ and Schwartz distributions.


