I want to prove that the convolution of a distribution is a $C^1$-function i.e. I want to prove that $u \ast \phi \in C^1$. I want to use the usual definition of derivative. Can anyone please help me with that.
Here`s what I have:
$$ \frac {u\ast \phi (x+te_i-a)-u\ast \phi (x-a)}{t} = \frac{\langle u,\phi(x+te_i-a)\rangle-\langle u,\phi (x-a) \rangle}{t} $$ $$= \langle u, \frac{\phi(x-(a-te_i))-\phi (x-a)}{-t} \rangle$$ and this converges to $$- \langle u,\frac { \partial \phi}{\partial x_i}(x-a) \rangle$$
then I have a note saying that we need to do it in a uniform way, show that the difference quotient converge to derivative in the topology of $\mathcal D (\mathbb R)$. I seriously don`t know what does this mean:(.
Thanks