= Convolution of a tempered distribution with a Schwartz function
{title2=$T*\psi$}
For $T\in\mathcal S'$ and $\psi\in\mathcal S$, define $(T*\psi)(x)=\langle T_y,\psi(x-y)\rangle$. Translations are smooth in the <Schwartz space>, so every derivative is $\partial^\alpha(T*\psi)(x)=\langle T,\partial^\alpha\psi(x-\cdot)\rangle$. The finite-seminorm continuity estimate for $T$ implies $|\partial^\alpha(T*\psi)(x)|\leq C_\alpha(1+|x|)^m$ for one $m$. Thus the result is a <smooth function> defining a <tempered distribution>, but it need not be a <Schwartz function>: $1*\psi=1$ when $\int\psi=1$. If both factors are radial, testing the rotation action on the translated $\psi$ shows that the convolution is radial.
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