Let $T \in \mathscr{S}'(\mathbb{R}^n)$ and $f \in \mathscr{S}(\mathbb{R}^n)$. We may define their convolution as $$(T * f)(\varphi) = T(\tilde{f}*\varphi)$$ where $\tilde{f}(x) = f(-x)$. The above convolution is defined to be a tempered distribution, that is $(T * f) \in \mathscr{S}'(\mathbb{R}^n)$. In Reed and Simon (Volume II, Theorem IX.4), they say
$T*f$ is a polynomially bounded $C^\infty$ function, i.e. $T * f \in O_M^n$. In fact, $(T * f)(y) = T(\tilde{f}_y)$ and $$D^\beta(T * f) = (D^\beta T) * f = T * D^\beta f.$$
Here $\tilde{f}_y$ is the function $f(y-x)$.
I am confused as I thought $(T*f)$ is a distribution, but in the above passage they are treating it as a Schwartz function by applying it to $y$ and bounding it by a polynomial. How would this make sense for a distribution?