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 $S^\prime$ 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.

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.