I was wandering if there are any theorems/ideas that would help me with the following situation (I came up with it myself and have been unable to find anything whatsoever).
Say you have two continuous functions $\;f,g : \mathbb{R}\to \mathbb{R}$ such that
$\displaystyle\sum_{t=1}^\infty f(t) = \sum_{t=1}^\infty f(t)g(t)$
Is there anything that can be said about the relationship between $f$ and $g$?
Trying to figure this out on my own has just led to me going around in circles. The idea of taking the integral of both occurred to me, but that seems to introduce too big of an error term. Also, I tried regarding $\;f$ and $g$ as vectors
i.e. $\;\;\;\vec{f}_2 = f(2)$
and multiplying them by the infinite-dimensional identity matrix $E$...then considering
trace$(E\vec{f}) = $ trace$(E\vec{f}\vec{g})$
But this doesn't seem to introduce anything new. I'm at a loss as to whether this is too general of an idea or if I'm missing something critical here. Any help would be greatly appreciated.