For simple Heaviside functions we see that $H(x - 1)$ describes $$H(x - 1) = \left\{\begin{array}{ll} 1 & x > 1 \\ 0 & x < 1 \end{array}\right. \hspace{5ex}$$
So for a function of $H(\pi^2 - 4x^2)$ would you have to set the argument equal to zero and find x as before and get:$$H(\pi^2 - 4x^2) = \left\{\begin{array}{ll} 1 & x > \pi /2 \\ 0 & x < \pi /2 \end{array}\right. \hspace{5ex}?$$
If anyone has any ideas, that would be great!
Besides, the two different cases in the second equation are exact.
One can consider $H(f(x))$ as the characteristic function of the set
$$F = \{x \ | \ f(x)>0 \}$$
Therefore $H(-f(x))$ is the characteristic function of the complementary set $\overline{F}.$
In the same vein : $H(f(x)).H(g(x))$ is the characteristic function of the intersection of two sets, etc.