i am learning about distributions and distributional derivatives, and would like to check my understanding.
suppose we have a step function that jumps by a unit at every integer:
$$H(x)=
\begin{cases}
 \lfloor x\rfloor, & x>0; \\
 0, & x<0; 
\end{cases} $$
my goal (just for understanding) is to take the distributional derivative and return something that looks like a dirac comb- a delta at each integer.

Question 1. In order to create a distribution we can simply convolve, $H(x)$ against a smooth, compactly supported test function, $\phi(x)$,
$$ H(x)\*\phi(x)$$, where $\*$ is the convolution operator.
Is this a valid operation here? most of the material i've found on convolution deals with two functions with compact support. i read elsewhere that as long as $\phi(x)$ has compact support and $H(x)$ is locally integrable, then the convolution is valid - i haven't seen a proof of this.

Question 2. To obtain a distributional derivative, can we simply convolve $H(x)$ with $\phi'(x)$?
$$ H(x)\*\phi'(x) $$

thank you