In the theory of automorphic forms and multiple Dirichlet series, we often take inverse Mellin transforms of Dirichlet series to come up with Tauberian theorems, like the Ikehara Tauberian method. In particular, from

$ \displaystyle \sum_n \frac{a(n)}{n^s}$, we might expect to use Mellin to get asymptotics for $\displaystyle \sum_{n \leq x} a(n)$.

Sometimes, the series doesn't behave well enough to integrate. A common way around that seems to be to multiply by the gamma function to get better decay. But this has the side effect of 'smoothing' the sum to get asymptotics for sums that look like $\displaystyle \sum_{n \leq x} a(n) e^{-n/x}$.

For a while, I just accepted that sometimes we need to accept smoothed sums. But I began to wonder how hard it would be to recover the regular, un-smoothed sum. So I'm prompted to ask:

Suppose we can calculate $F(x) := \displaystyle \sum_{n \leq x} a(n) e^{-n/x}$ for any given $x > 1$. For a fixed $N$, can one recover $\displaystyle \sigma_N := \sum_{n \leq N} a(n)$ from $F(x)$? In particular, it is ever possible to recover $\sigma_N$ by calculating $F(x_i)$ for a suitably chosen, but

finiteset of $x_i \in \mathbb{R}$?