Dirac Comb Times Step Function

Question

Can someone explain me what is the effect of the Heaviside step function $\Theta(t)$ on a Dirac Comb (Fourier series)?

$$ \left[\,\,\sum_{n=-\infty}^{\infty}c_{n}\,\delta\left(t - nT_{0}\right) \right]\Theta(t) = ?$$

Answer 1

Assuming that $T_0>0$ then drop all the -ve n from the summation. Care needed with n=0, I think the Heaviside(t=0) is equal 1/2 in which case you will have a term $c_0 /2$. +n remains unchanged.

Answer 2

It is dangerous to try to truncate a Dirac comb so "sharply" as by multiplication by a translate of Heaviside. It is safe only when the jump in the translate of Heaviside is strictly between the spikes of the comb.

(It is dangerous to think that "declaring" the value of Heaviside at $0$ to be $1/2$ repairs anything or makes it safe.)