Let $u$ be a tempered distribution and $\psi$ a Schwartz function.
I have seen two definitions for $\psi*u:$
- $\psi*u:$ defined as a function of $\mathbb{R}^d,$ that is, for $x \in \mathbb{R}^d,\psi*u(x)=u(\psi(\cdot-x)).$
- $\psi*u$ defined as a tempered distribution, that is, for $\phi \in \mathcal{S},(\psi*u)(\phi)=u((\psi(-\cdot))*\phi).$
Also among the properties of convolution by tempered distribution, we have for $\psi_1 \in \mathcal{S},(\psi_1*\psi)*u=\psi_1*(\psi*u).$
- How to we usually distinguish between definition 1. and 2. ? Are they equivalent? How?
- The relation $\psi_1 \in \mathcal{S},(\psi_1*\psi)*u=\psi_1*(\psi*u),$ does it hold in the sense of definition 1. or 2.? Or is it both?