I would like to study the exponential sum in an appropriate functional space.
In particular:
$f(x): \mathbb{R} \to \mathbb{R} $
$f(x) = \chi_K \sum_{i=1}^{M} R_i \exp{(s_i x)}$
where $R_i, s_i \in \mathbb{C} \forall i$, $M$ is finite and $K$ is a compact interval on $\mathbb{R}$.
I would like to know which functional space is span by this set of functions.
Is it possible to add some restrictions, or generalizations, in order to have a good closure for the functional space (i.e. Banach, Hilbert, etc...)?
References are really appreciated.