Does the convolution $q(t) = q'(t)\circledast\theta(t)$ always holds? Why?, if not, What is needed?

Question

Does the convolution $q(t) = q'(t)\circledast\theta(t)$ always holds? Why?, if not, What is needed?

$\theta(t)$ is the standard unitary step function. I want to know which characteristics has to have $q(t)$ to make $q(t) = q'(t)\circledast\theta(t)$ true... and if true, try to understand why it is so, and if it can be think of the derivatives as the step-response-function of LIT systems (Linear and Time-Invariant).

Answer 1

Taking the bilateral Laplace transform we have

$$ Q(s) = (s Q(s))\frac 1s $$

so

$$ Q(s)=Q(s) $$

which is true for all $|q(t)|\le M e^{\alpha t}$