Inspired by this question, I started to think about which functions $t\to f(t)$ have the convolutional property ($*$ being convolution) : $$\lim_{t\to \infty}\frac{(f*h)(t)}{(f*b)(t)} = 2$$
Where $h$ is Heaviside step-fun and $b$ is an infinite box-train of some box width $\Delta_t$ multiplied with $h$.
Just for context the mentioned question is an example where a particular choice of partial sum gives us a discrete version of this where $f(t) = t^2\cdot h(t)$ and $\Delta_t = 1$
How to derive which functions this is true for?