Let $\mathscr{S}(\mathbb{R})$ denote Schwartz space, $T \in \mathscr{S}'(\mathbb{R})$, and $f \in \mathscr{S}(\mathbb{R})$. How can we make sense of the product $fT$?
In the second volume of Reed & Simon's book on mathematical physics, they give the following theorem:
Let $T \in \mathscr{S}'(\mathbb{R})$ and $f \in \mathscr{S}(\mathbb{R})$. Then $\widehat{fT} \in O_M^n$ (the set of polynomially bounded $C^\infty$ functions) and $\widehat{fT}(k) = (2\pi)^{-n/2}T(fe^{-ik \cdot x}).$
In their proof, they say that using the result $$\widehat{T * f} = (2\pi)^{n/2}\hat{f}\hat{T} \tag{1}$$ and the Fourier inversion formula that $$\widehat{fT} = (2\pi)^{-n/2}\hat{f} * \hat{T}. \tag{2}$$
Using the Fourier inversion formula on (1) as they suggested gives: $$T * f = (2\pi)^{n/2}\check{\hat{f}\hat{T}}$$ but this is not the same as (2). How did they obtain this?
Secondly, how does one interpret the product $fT$ if the two have different domains? There is a result that says the convolution $T * f$ can be interpreted as a $C^\infty$ function, but I am not aware of anything similar where the convolution is now an product.