Suppose I have two functions continuous functions $f,g \in \mathcal{S}(\Bbb{R})$, the Schwartz space. Now I know that the following multiplicative formula holds. Namely, if $\hat{g}$ denotes the Fourier transform of $g$ and similarly for $f$, we have that
$\int_{\Bbb{R}} f(x)\hat{g}(x)dx = \int_{\Bbb{R}} \hat{f}(y)g(y) dy.$
Now I am reading the proof of this fact in Stein and Shakarchi volume 1 and it seems that we do not even need $f,g$ to be in the Schwartz space. Them being of moderate decrease is enough, namely there is a constant $A$ such that
$|f(x)| \leq \frac{A}{1+x^2}$
and similarly for $g$. Am I right in saying this?
Thanks.