How to find out the laplace transform of the product of two functions? How to find out the laplace transform of 
$$f(t) \cdot \sum_{i=1}^\infty a_i g_i(t),$$ 
w.r.t the variable $t$ on the domain $[0,\infty)$,  where $a_i$'s are constants with value $0\le a_i\le1$.
 A: In general, if $F(s) = \mathcal{L} \left\{ f(t) \right\}$ and $G(s) = \mathcal{L} \left\{ g(t) \right\}$, the Laplace transform of the product can be found in that way:
$$\mathcal{L} \left\{ f(t)g(t) \right\} = \frac{1}{2\pi i}\lim_{T \to \infty} \int_{c -iT}^{c+iT} F(\sigma)G(s-\sigma)d\sigma = \frac{1}{2\pi i} \lim_{T \to \infty} \int_{c -iT}^{c+iT} F(s - \sigma)G(\sigma)d\sigma ,$$
where $c$ is any constant such that $ c + ia $ is in the region of convergence of F for every $a \in \mathbb{R}$. 
But I think I'm not trully understanding your question. Have you said that $f,g_i$ are constant functions on their domain? Then you can express them as linear combinations of heavyside step functions and their Laplace transform will be easier to find. 
A: Then, if they are just continuous functions, I only see 2 ways to get the Laplace transform of $h(t) = f(t) \sum_{i=1}^{\infty} a_i g_i(t) $:


*

*Considering $F(s) = \mathcal{L}\left\{ f(t) \right\}$, $G_i(s) =\mathcal{L}\left\{ g_i(t) \right\}$ for each $i \in \mathbb{N}$. Since the Laplace transform is a lineal operator, for every $n \in \mathbb{N}$ we have that $\mathcal{L}\left\{ \sum_{i=1}^n a_i g_i(t)\right\} = \sum_{i=1}^n a_i G_i(t)$. I'm not sure if there is any problem with linear combinations of infinitely many functions, so you may prove that $\mathcal{L}\left\{ \sum_{i=1}^{\infty} a_i g_i(t)\right\} = \sum_{i=1}^{\infty} a_i G_i(t)$ (as a limit, at least). And then, use the rule I gave you in the other answer.

*If you don't need the Laplace transform of $h(t)$ to depend on the Laplace transforms of $f$ and $g_i$ and you have more information about them, try to do it as a unique function. But for this you need to know explicitely (or at least, "better"), $f(t)$ and the $g_i(t)$ (or directly $h(t)$). 
I'm working in a similar problem and I used the power series developement of one of the functions. Since $\mathcal{L}\left\{ t^n f(t) \right\} = (-1)^n \frac{d^n}{ds^n}\mathcal{L}\left\{ f(t) \right\}$, I obtained the Laplace transform of the product as a series on the derivatives of the Laplace transform of one of the functions in the product. But I don't think this is a good way. 
I hope it has been helpfull!
